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Another response has linked to some excellent discussions of our knowledge of Tiwanaku society overall, so I'll try and address three topics you've brought up here:

1. Tiwanaku masonry
2. The Lake Titicaca shoreline
3. Tiwanaku's position relative to Andean society and history - that is, I'll question the idea that it was an unexplainably great or advanced society relative to its place in its Andean contexts

Before all of that, I'll begin with a caveat: archaeologists are aware of and struggling with how little we know about Tiwanaku. As is the case with so much of Pre-Columbian history in the Americas, the loss of information due to time is exacerbated by centuries of colonization, destruction, discrimination, and erasure that continues to this day. Even generalities about Tiwanaku (such as how powerful and expansive of a state it was) are often heavily debated. Now, let's get to the topics I listed.

Tiwanaku Stonework

Tiwanaku and Puma Punku have utterly fantastic stonework. The detail, precision, scale, and so many other factors of the sites' stone art is truly remarkable. However, there is good reason to believe that its creation was not impossibly advanced for the times and places in which archaeology generally recognizes it thrived in (that is, somewhere around 0 AD to 1000 AD in the central Andes, with a couple hundred years of range on those start and end dates).

If you are truly invested in exploring this topic I highly recommend reading through The Stones of Tiahuanaco by Jean-Pierre Protzen and Stella Nair. The version I've just linked should be publicly accessible. For now, though, I'll just focus on the authors' archaeological experiments that successfully reproduced Tiwanaku-style stonemasonry with tools such as stone scrapers and hammerstones. That section begins on page 154 of the book I linked, and I'll quote page 162:

Without a doubt, the experiment showed that motifs, like those found at Tiahuanaco, could be carved with simple stone tools (Figure 5.14). No fancy theory is needed to explain Tiahuanaco stone carving. Using these stone tools, Nair was able to closely reproduce what the Tiahuanaco accomplished: dimensional precision, right angles, and sharp edges and corners on both the interior and exterior of the motifs.

The authors did have some trouble reproducing the "perfect planarity" of Tiwanaku stonework in their experiment's carved motif's interior planes. However, they point out how their (incorrect) assumption of being able to flatten interior surface after creating the motif is not a death knell for the viability of such techniques on page 164:

our subsequent field analysis revealed that...a flat surface was made to define a motif in one stage of the process, only to be subse-quently destroyed and then remade again later, a few millimeters deeper into the stone block... Tiahuanaco stone masons approached the desired final surface gradually by removing one layer of excess stone after another, with ever more care the closer they came to the target surface...very early in the process, [stonemasons] created carefully leveled sections of the finished surface to serve as references to the final surface. This is consistent with the earlier reported observations that rough work and finishing touches were made simultaneously on one and the same stone

I'd like to add a personal note that I think that using obsidian and flint polishers and especially incision blades is incredibly cool. Aside from the experimental reproductions, I'll highlight the reality of mistakes in Tiwanaku stonework that point to their construction methods. For example, Nair and Protzen note

a clear view of an incision line can be seen in a small stone fragment at Pumapunku, which reveals a rare mistake...an incisor blade has overrun the interior corner, providing us with an unobstructed view of a Tiahuanaco incisor mark (Figure 5.25)

A final discussion topic for the plausibility of constructing Tiwanaku is the question of transporting the sometimes massive blocks that were involved in Tiwanaku work. There is good evidence that these blocks were often dragged directly over land, without wheels or sleds. Figure 6.6 on page 180 of Nair and Protzen's book is excellent proof - it shows a block in Tiwanaku that still has parallel marks from being dragged over land.

Of course, land isn't the only means of transport, which brings us to water and...

The Lake Titicaca Shoreline

I am not a hydrologist, geologist, or someone with otherwise direct detailed knowledge of Lake Titicaca's water levels and depths. If someone with that knowledge does see this post or response, I hope they pitch in. However, there is archaeological evidence that suggests Tiwanaku was built during a time when the lake was not at the city's level, yet water transportation was still a plausible (and perhaps probable) method of moving some stones.

This evidence comes through the existence of Iwawi. I can't find many maps showing its location, but this paper shows it on page 105, figure 18 (and also lists it as a Tiwanaku society burial site). Iwawi is located on the shores of Lake Titicaca, around 20km from central Tiwanaku city. A direct road from Iwawi to Tiwanaku has been discovered, and even more tellingly, multiple andesite blocks have been discovered around Iwawi.^(1) Pre-Hispanic societies along lake Titicaca are known to have built large reed boats and rafts (most famously, in the form of Uros "floating islands" that still exist today). Additionally, The Stones of Tiahuanaco mentions contemporary experiments which were able to transport a 9 metric ton stone on a traditional Andean reed boat. Notably, these boats are smaller than the reed floating island rafts that can theoretically carry even larger stones.

Iwawi working as a port for Tiwanaku's construction also makes sense when we consider how

1. the closest source for Tiwanaku's limestone comes from areas across lake waters
2. Tiwanaku's andesite has trace elements suggesting it came from a location across Titicaca's waters.^(2)

It seems like at least some stones were quarried in distant places, dragged to the shore, floated across the lake to Iwawi, and then dragged from Iwawi to Tiwanaku proper. If this is true, it suggests that the lakeshore was at Iwawi's level when Tiwanaku was built.

Tiwanaku's Position in Andean Space and History

I hope I'm not putting words in your mouth, but it seems like you find archaeological academia's general agreements about Tiwanaku suspicious at least partially because of a belief that Tiwanaku was unexplainably advanced and great given its time and place. There are points to be made about the problems with understanding history, technology, and society as a linear thing that advances or regresses. I'll link these previous responses as a starter on that topic. But I think that the heart of your question is an idea that the buildings and characteristics of Tiwanaku don't make sense in the specific position within the flow of Andean history where archaeologists have located that society. That is, a kind of trifecta of: Tiwanaku was too complex and advanced to have existed right after the societies that archaeologists say came before it, known societies that were supposedly contemporaneous with Tiwanaku are incomparable, and information about societies after Tiwanaku does not make sense given Tiwanaku's advanced nature. To address that trifecta, let's take a look at Tiwanaku's position in those three stages: before Tiwanaku, contemporaneous with Tiwanaku, and after Tiwanaku.

I've written about the Tiwanaku culture in several previous answers:

• On monumental architecture
• On claims of monotheism
• On cosmopolitanism in the capital

This should fix your timeframe, moving it into the first millennium AD, and provide the basics on temples such as Puma Punku in the Tiwanaku capital. There's much, much more to be said of course. The city of Tiwanaku gets a lot of press, but it was just the center of a cultural/religious phenomenon from Peru to Chile.

Do you have any more specific questions?

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This response does feel incomplete without addressing the language and assumptions in your question.

Through no fault of your own, you've stumbled upon some sensationalist claims about the date of the Tiwanaku culture and the nature of Andean stoneworking. This is surprisingly easy to do. Even in generally respectable popular resources, Tiwanaku is often singled out as "mysterious." Take this Great Courses series in which all the other cultures get to have gold and pyramids and mummies but Tiwanaku is simply "enigmatic." Between the early travelogues of G. E. Squier and the explicitly racist early "scholarship" of Arthur Posnansky, Tiwanaku has long been portrayed as mysterious and unfathomably ancient in the popular imagination. In the present, the tourism industry takes much of the blame; TripAdvisor has gone ahead and slapped "one of the most mysterious ancient ruins in South America" on the site's page. The rest of the blame is on dudes who publish crap like:

My only concern is academia being too dogmatic and set in their own beliefs to follow the clues

This is a common line used by fraudsters to sell more books and clicks. Their claims (if one can call them that, for they are rarely ever definite or specific) cite such fragmentary evidence that the authors need to establish themselves as radical outsiders revealing hidden truths to add appeal. Little evidence is likewise given for the supposed dogma of archaeologists, outside of "they told me I was wrong" or "they won't accept my argument until they see the evidence."

Unfortunately it's also a very convincing line because their audience has no understanding of the extent of the archaeological evidence, and their ideological bents make them not terribly interested in getting to know it. Not only do these hacks know this, they rely on it. I've written here about how certain authors actively lie about what archaeologists do/don't say and know. /u/mictlantecuhtli has observed here that Andean masonry traditions are easily misconstrued as impossibly complex because so few people alive today know anything about stoneworking. Since you mention it, the Younger Dryas Impact theory has similarly functioned as a sort of "black box" for all these claims that hand waves the lack of evidence with a cataclysmic event that oh-so-conveniently destroyed all of it.

What do I mean by "the extent of the archaeological evidence?"

Pop archaeology likes to think in terms of Big Finds and Cool Artifacts. It assumes our knowledge of the past comes largely from singular impactful discoveries: much as Newton discovered gravity and revolutionized physics, Bingham discovered Machu Picchu and revolutionized Andean archaeology. This fits more snuggly with traditional understanding of the history of science, and it's much easier to package as a press release. It's a sort of Duning-Kruger effect wherein people assume that, for instance, descriptions of each object posted on /r/artefactporn is based on analysis of that specific object, rather than on the analysis of the 100s of other similar artifacts. In this model, there very well could be a trail of "clues to follow" that lead to The Next Big thing. If archaeologists are basing their claims on the handful of sites and artifacts familiar to the public, all it takes is just one artifact, one site, one fascinating insight to force them to re-analyze everything.

However, archaeological progress doesn't happen via individual discoveries, it happens at the confluence of many, many small data points. Machu Picchu is undoubtedly cool, but there's very little that has come research at the site that might be said to, on its own, "advance our understanding of the Inca Empire as a whole." Meaningful interpretations have come when comparing what we know about other Inca sites with this one. Context is the key word in archaeology, and that applies to individual artifacts and to entire sites. One Moche portrait can't tell us much. But hundreds? That's when we can start using patterns to make claims. Claims are built on the relationship between objects and their locations- never on any one object.

Let's apply this to something very basic: when was Tiwanaku built?

The quick and easy answer is to use radiocarbon dates. Thankfully, we have plenty. This recent article ran Bayesian analysis on 60 samples from the period immediately before and during the construction of Tiwanaku and contemporary sites. This is but a selection, of course; there are dozens and dozens more radiocarbon dates from Tiwanaku. They all point to a the "classic" Tiwanaku state beginning just before 600 AD.

We might also look to stylistic seriation. Tiwanaku is known for its characteristic ceramics, so much so that it formed the backbone of the earliest pan-Andean chronologies. It did not come from nowhere. The local Qeya style predates the classic Tiwanaku style by some 150 years and included forms that are clearly early versions of later standards. Compare, for instance, letter C in this figure with a typical Tiwanaku incensario: the feline head, scalloped edges, and even the layout of imagery on the body are a clear antecedent.

The most foundational tool for answering is stratigraphy: old things are found beneath new ones. I've discussed this in detail in that first linked answer, but many of the structures standing at Tiwanaku today are renovations to earlier buildings or built directly on top of them. There's an immediate spatial relationship. The first Tiwanaku temples appropriate the prior generation's important spaces, while later constructions are more directly assertive about the new tradition.

Lastly, the influence of Tiwanaku extended far beyond the city itself. The chronology of the heartland lines up with dates from affiliated centers in the Amazon foothills, with trade centers in the Chilean desert, and with migrant colonies established on the border with the Wari state on the Peruvian coast. Tiwanaku presence at sites across the south-central Andes must also fit in with local stylistic evolution, with radiocarbon dates, and with stratigraphy.

And you know what? It all fits together. There's literally hundreds of thousands of data points that point to Tiwanaku's emergence and expansion in the 6th and 7th centuries, derived from a dozen different methodologies. The lowest layers have the earliest dates and the antecedent styles and this is true across the region. I can't shift the date of Tiwanaku as I please without having to account for an extraordinary number of resultant issues.

This is why archaeologists will appear dismissive of "clues," no matter how exciting they might seem individually to you. When you have an observation with multiple potential causes, you must determine the cause contextually. Claims that Tiwanaku is millennia old, such as "Puma Punku is on the ancient shore of Titicaca" or "Akapana lines up with the way stars were 16000 years ago," are workable when everything exists in isolation. But every data point is tangled up in so many other ones that any single observation must be interpreted in terms of them. Is Puma Punku where it is because of ancient lake levels? Or is their a confounding variable (namely, the natural ridge it sits on) that would make that spot both a likely place for a lake to end and a nice place to build a temple?

Archaeological claims are like a Lego model. If a piece doesn't fit, you don't ask if the rest of your build is wrong, you flip the piece around.

While there's always more that can be said, u/Commodorecoco has covered Tiwanaku pretty extensively in previous answers, including this one focused specifically on Puma Punku.