TL;DR: Complex civilisations did exist, particularly on the edges of Amazonia. They have not been lost, although many were destroyed by colonialism and the diseases brought with it.

It's a great question, but perhaps best answered by breaking your question down into a few parts:

Is it possible that the Amazon was less densely forested when humans first arrived?

The size of the Amazon has altered over time, but when humans arrived - around 11,000 years ago - forest cover was broadly similar. This is measured by the analysis of pollen counts in cores drawn from lake and river beds. So no, early Amazonians encountered a broadly similar biome to what we see today.

Could forest growth obscure traces of civilisations?

Yes, absolutely. Archaeology is hard enough anywhere, let alone in the middle of a rainforest! Amazonia is a vast region, so even picking where to dig is a challenge. Investigation has tended to stick to river banks and sites visible from the air. The archaeologist Anna Roosevelt uncovered new sites simply by digging deeper - into layers of earth covered before archeologists believed humans had arrived.

That being said, archaeologists aren't just digging randomly. They're following a sensible, logical approach - the proof of which is that they have found evidence of older and more complex civilisations than were thought to exist. It's very hard for a major civilisation to disappear without trace - complex societies produce trade goods, settle on multiple sites and have a profound impact on their local and regional environments.

Could there be a lost civilisation more complex than anything found to date?

Almost certainly not. We can be confident of this because the complex societies that existed in Amazonia have left significant records. The Marajoara society, based on the Amazon delta island of Marajó, left a wealth of pottery at multiple sites, as did other pottery-producing cultures along the river in places such as Santarem. Geographic evidence of agriculture has also been found - earthworks allowing terraced farming, waterworks demonstrating fish farming and analysis of soil and trees surrounding villages that demonstrates human impact on the environment.

These findings tally with eyewitness accounts at the time of early European colonisation. Large riverbank settlements did exist, with populations in the low thousands, perhaps with a few regional settlements in the high thousands. Most settlements existed on the fringes of the rainforest, however, where tree cover was lower and agriculture was easier. Within the forest, the difficulty of clearing and maintaining land, combined with the relative weakness of the soil, would make a large city impossible.

This is reinforced by the history of contact - as the Amazon has been explored, many societies have been encountered but none so large or so centralised as to challenge existing theories.

What about the new discoveries?

The problem with reporting on a lot of these 'new discoveries' in the Amazon is that they're not as new or challenging as you might think. The so-called 'sistine chapel of the Amazon' in Colombia is definitely a remarkable and impressive site, but its discovery was sensationalised to promote a television series. Amazonian rock art is well-studied and this site, although large, fits very well with theories of Amazonian settlement.

Archaeological/historical knowledge is relatively confident in the existence of relatively complex societies on the edges of the rainforest (complex in that they harnessed specialised labour and large-scale agriculture) as well as smaller societies along river banks. Even deep within the forest, settlements existed, practicing agriculture in forest clearings. These settlements, however, were more limited in size due to the low quality of soil and the difficulty in maintaining clearings.

There are active debates - the starting date of human settlement in the Amazon may be earlier than we think, although the evidence is quite limited for now. The pre-colonial population of the Amazon is also subject to some speculation. But the new discoveries you mention are not particularly explosive to existing academic theories - they are similar to earlier discoveries and support current thinking.

So why are they presented as dramatic?

Well, there are a mix of reasons. Most people's views of the Amazon are of an untouched wilderness populated by small groups of indigenous hunter-gatherers. Myths of hidden cities in the jungle - most notably the Lost City of Z and El Dorado - have gripped popular imagination since the early 20th century, if not earlier. Politically, the exploitation of the Amazon is linked to a belief in its emptiness - during the Brazilian dictatorship, settlement of the Amazon was promoted under the slogan 'Land without people for people without land'. And, inevitably, media like to hype up the novelty of discoveries to drive engagement.

But within the disciplines of history and archaeology, there is a sufficient body of knowledge to explain why these discoveries exist. Complex civilisations did exist, particularly on the edges of Amazonia. They have not been lost, although many were destroyed by colonialism and the diseases brought with it.

Source: J. Hemmings, River of Trees, (Thames and Hudson, 2008)

First let's clear some things up.

As discussed here by myself and several others, the so-called "Sistine Chapel" has received significant press only because of publicity for a then-upcoming BBC documentary. The site was neither new to archaeologists nor did it significantly alter our understanding of the region at that time. The scale, of course, is impressive. But it's an early Holocene site from thousands and thousands of years before the cultures which produced earthworks.

Also, "the Amazon" encompasses an enormous area with diverse biomes. Many of the stories about Amazonian archaeology that get picked up by the popular press, such as this one, describe finds from the western edges of Amazonia where dense forest gives way to flat, flooded savannas dotted by clusters of wooded "islands." Much research has been done in the Llanos de Mojos region of Bolivia (the yellow and green hashed area on this map). Some estimates put over 100,000 people in the region at its pre-contact peak, and it is criss-crossed by canals, causeways, and residential platforms that made living there possible. The question is somewhat moot in this case, though, since these constructions are and have always been uncovered by forest.

As for your question:

As discussed here by /u/Qhapaqocha and I, the act of searching for and distinguishing "civilizations" among other types of societies is as arbitrary as it is political. The category doesn't reflect any meaningful social reality; those who argue that something must be a civilization usually have something else to gain from that designation.

However, this isn't quite the way in which you are using it. "Civilization" as short-hand for "sedentary, stratified society" is pretty common. There are still many archaeologists who find designations like "chiefdom" and "complex" useful when grouping societies for comparative studies. Though I'm more in the camp that considers these political classifications "archaeological delusions", folks who still use them know what they're doing.

Problems begin when we classify a society through these analytical categories, then try to find evidence for other aspects we would expect the society to have based on those, or try to come up with reasons why we can't find those aspects. It's a deductive logic at odds with inductive research (which I've extrapolated on here). We can't, for example, find expansive irrigated fields, claim that whoever built them must have been a chiefdom, then go look for other things we might associate with a "chiefdom." Anthropologists made up the term to describe certain societies they observed, and then others spent the 20th-century showing just how insufficient it was to account for the variation in sociopolitical structures across the globe. It's circular reasoning to expect a society to have certain traits because we called it a chiefdom.

Thus, there's no real answer to your question, in the sense that we can't use the expansive constructions that we do know about (i.e. over 200 sites with earthworks just in the western Amazon) to say much about those that we don't know about. Rather than go on with simple descriptions of what we know about Amazonian agriculture and earthworks, it seems like the better answer to your question is how we know those things.

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Like most South American archaeology, Amazonian archaeology began in the early 20th-century. The earliest formal excavations were completed in 1908 by Erland Nordenskiöld, followed by smaller scale operations by fellow Swede Stig Ryden. These excavations focused on a handful of ceremonial mounds rising up from the flat savanna of the Llanos de Mojos in Bolivia. Given the colonial Spanish accounts of Gaspar de Carvajal and others, it was no surprise that there were such constructions. Of course, colonial documents can rarely be taken at face value. The question has never been if sedentary agricultural societies lived in the Amazon, but when they lived there and who they were. On the opposite side of the Amazon, Betty Meggers and Clifford Evans began excavating at Marajó Island on the Atlantic coast of Brazil in the 1940s. Like Mojos, the island features numerous ceremonial and residential mounds.

These first studies focused on singular architectural features, burials, and the ceramics and tools found within them. Many prominent Americanists offered narratives to fit the finds within the (limited) archaeological landscape. Julian Steward's version would become the standard during the '50s and '60s, thanks to strong support from Meggers. He argued that the soils of Amazonia were too poor to support substantial agriculture; the monumental architecutre in Mojos and Majaro were the result of more "civlized" peoples from the Andes that migrated east along the Amazon. The evidence for this was limited, mostly built on some supposed similarities between Mojos ceramics and that of Tiwanaku in the Bolivian highlands- though this was debated by those who had worked at Tiwanaku like Wendell Bennet. More impactful was Steward's extreme neo-evolutionary views. He understood human societies to be fundamentally dependent on their environments, and it only made sense that Amazonia, which was either constantly flooding or densely forested, would not develop social complexity on its own.

Steward and Meggers theory would quickly lose relevance during the '60s, thanks mostly to William Denevan's pioneering use of aerial photography in Llanos de Mojos. Aerial photography had been used in archaeology before, but it was uncommon at the time for an archaeologist to specifically hire a helicopter for photographic survey. His full 1966 report from surveys in 1962 can be downloaded here. Both Denevan at the time and archaeologists today recognize how essential the aerial survey was for understanding the nature and extent of earthworks in Mojos. Nordenskiold and others weren't exactly looking for landscape-scale features, but even if they had been, they would be very difficult to map from the surface. The massive amount of causeways, canals, and agricultural features in Mojos suggested a large number of residents who supported themselves without reliance on Andean societies.

Modern studies have discovered architectural features deeper in the forested Amazon basin through the use of LiDAR. LDAR is able to detect surface contours through foliage cover, which Denevan had identified as the main obstacle to further progress via aerial photography alone. Denise Schaan's work in the Santrem region of eastern Brazil has used LiDAR survey to reveal a regional hierarchy of sites- single large settlements surrounded by several smaller ones. Such patterns have traditionally been interpreted as a sign of a stratified political system.

Recent research has also clarified the timeline of occupation and abandonment in Amazonian sites, as well at their agricultural practices, through improved soil analysis techniques. Hot environments that are constantly being flooded are not the best for preserving artifacts, and so archaeologists have had to turn to other sources of information. Analysis of pollen and phytoliths produced by plants and preserved in soils tell us that manioc was first domesticated in the Amazon. Higher resolution radiocarbon dates from better isolated soil samples suggest that many sites in Mojos were abandoned before either the Inca or Spanish reached the region. Chemical characterization of terra petra soils (described in another comment here) has allowed us to differentiated anthropogenic (i.i formed by human activity) soils from naturally occurring ones and to better guess at the exact practices that produced it.

So, to summarize, while deforestation has certainly revealed many structures that would otherwise be covered, there existence has never been "unknown" to archaeologists. Advancements in our understanding of Amazonian societies have instead been tied to technological developments and to abandoning clunky theoretical approaches.