/u/400-rabbits has answered a version of this question with punny horticultural flair before.

EDIT: /u/Snapshot52 has an answer about a North American "cradle of civilization" with links galore.

I've never seen this list of cities in the Americas list before. I have some reservations about calling sites like Aspero or Caral cities. I'm less familiar with central America, but the Andean cases are suspect. Anyway, the definition of a city is not straightforward or easy, so there's loads of room for debate on that matter. Michael Smith, an expert on urban archaeology, sums it up well in a series of definitions posts. In his latest one he basically says screw definitions and you know a city when you see one.

But there's another issue here: you say you are looking at a list of earliest settlements, but it is a list of earliest cities. Ignoring what I just said about this list, if we accept that all of these are indeed cities, that still is a tiny sample of all settlements in the Americas. Urban living is a very specific type of settlement, and comes into existence as a result of various historical, demographic, and economic reasons.

Most settlements are not cities. Urban living is earlier in Central and South America than it is in North America for various reasons (worthy of another question post). But the key point here is that cities are not a direct outcome of time spent in a region, but a product of specific developments. Smith discusses this in many of his papers, but his 2002 article is a good overview.

So now that we've gotten the cities and semantics aspect out of the way, let's look at your question. Essentially, we need to look at two things: how old are the oldest sites in South America, and how old are the oldest ones in North America?

South America

As /u/Gus_Frin_g said, Monte Verde is the earliest confirmed settlement in South America. While 14,500 years ago has been an accepted date, Dillehay has always said it is older. Dillehay et al. (2015) conclude that it was settled sometime between 15,000 - 20,000 years ago, and Perez et al. (2016) support a similar date. The Arroyo Seco site in Argentina was settled by at 14,000 years ago, suggesting that the southern cone of South America had a growing population by then.

What about the rest of South America? The Pedra Furada site in Brazil has long been argued to have some very ancient dates (as early as 30,000 years ago), but most archaeologists have not accepted this. More recently, new research says the site is at least older than 20,000 years old, but this is still not widely accepted. The main issue is whether the stones that they claim are tools are actually tools, or are natural stones that look tool-like (this sounds straightforward, but some stones stump even experienced stone tool analysts. I speak from experience here). There are actually a few sites in North America that have also been said to date to before 20,000 years ago, but Tune et al. (2018) have argued that all of these sites are entirely natural, and are not in fact archaeological sites.

On the other side of the continent, the site of Huaca Prieta in Peru has basically been continually occupied forever, with an early date of around 15,000 years ago.

Brandini et al. (2017) date the separation of various genetic lineages in South America. They conclude that migration into South America happened quickly after North America was first occupied, and they put 14,500 - 16,000 years ago as the likely first occupation of South America.

There is a lot of debate around any individual early find, but taken together we have good evidence for people in South America definitely by 14,500 years ago, maybe by 16,000 years ago, and mayyyyyyybe earlier. So we can use that as our date for earliest settlements of South America.

North America

Which brings us to your actual question: why aren't there older and more settlements in North America if people came from Beringia? Well, there are:

• Bluefish Caves, in Yukon, has evidence of human activity that dates 20,000 - 24,000 years ago. This was likely unrelated to later occupation of the continent (supporting an idea called the Beringian Standstill hypothesis).
• The Page-Ladson site in Florida is well-dated to 14,500 years ago (they collected 71 radiocarbon dates. With radiocarbon dates, more is better. 71 is a ton.)
• There's lots of debate around the Debra L. Friedkin site in Texas. Some papers argue that it is as old as 15,000 - 13,000 years old. Questions about the dating method (Optically-Stimulated Luminescence, or OSL) is the source of much of this debate, but it is a pretty reliable method.
• Nearby, the Gault Site is a well-known Clovis Period (13,500ish to 12,500ish) site. Last summer they found some artifacts below the Clovis layers. They date these layers to at least 16,000 years ago and suggest that people may have lived throughout the continent by then. This is a new find so it has not been subjected to much additional research, but they also used OSL to date these layers.
• Paisley Caves in Oregon has coprolites (preserved poo) that give the earliest human DNA in the Americas. The site is definitely pre-Clovis, and the coprolites have an estimated age of 14,500 - 16,000 years ago
• The Meadowcroft Rock Shelter in Pennsylvania was one of the first sites known to have evidence of Pre-Clovis occupation, and also the most contentious. There was a ton of debate about their dates, and whether they were accurate or contaminated. As far as I know this is not entirely settled, but the fact that there are so many other sites older than 13,000 years ago lends credence to Meadowcroft as well, and to the possibility of it dating 14,000 - 15,000 years ago. A couple of good short overviews are in this article. and in this one.
• One of the leading hypotheses for how people got into North America says that they travelled along the coast of Alaska, B.C., and Washington and moved inland south of the glaciers. There were various places along the coast, that may have been habitable, and many more submerged coastlines that are difficult or impossible to study archaeologically The other leading hypothesis (an inland ice-free corridor on the east side of the Rockies) is faltering but is not yet rejected.
• Michael O'Brien, who has worked on the archaeology of North America since the 70s, recently published an overview of this whole topic. He will go into a lot more detail than I can. He sees the earliest solid evidence of humans in North America at 14,500 - 15,300 years ago.

There are others. A lot of others, if we include all of the ones that are more suspect, but I have been avoiding doing real work for long enough so I'll wrap this up. The emerging picture shows that people came into North America (south of Alaska & the Yukon, at least) by around 16,000 years ago and spread rapidly throughout both North and South America. The Clovis period with its famous fluted points was a local innovation on an already populated continent.

North and South America were, in a sense, occupied simultaneously, at least at the scales that we can observe archaeologically (we typically have error ranges of a few centuries when looking at dates this old). With more data we should be able to get a more precise picture, but we always say that about everything we study.

Note: I've tried to find open-access articles or things that you can get on Researchgate (with an account). Let me know if you can't access a source and I'll try to find an open copy or a similar article that is open access.

Edits: tidying up wording and typos. May have missed some.

I'm sorry if this post violates the rules, but I'm wondering if there's any support from historians of the theory that the North American civilizations and settlements were wiped out by natural disasters (volcano eruption, ice age) or if this is still relegated to the realm of conspiracy theory.