Then of course their tools and other artefacts would have been traded all over the place. Stone tools from 100k years ago and more got traded or carried thousands of miles and all we see are implements for the hunter gatherer lifestyle.
It doesn’t make any sense.
For most of these cultures, "cities" did not exist, and even assuming they did, from the times of the first early-agricultural settlements, might have numbered a few thousand, perhaps tens of thousands of souls. With little transport, food was either sourced locally, or brought in on boats, rafts, or barges. Total collection areas for bulk goods of more than a few kilometers would be highly unlikely, though longer trade routes for rare materials (timber, metals, spices, hides) might have existed.
A principle relic of such settlements are their rubbish tips. Again, lack of transport meant that waste and filth were moved a few meters, for as long as the settlement existing -- decades, centuries, or millennia. In many places these are the principle evidence of early settlements, particularly shell mounds (shellfish are an easy, reliable, and sustaining prey), and can themselves measure several meters high and tens of meters across, particularly on otherwise flat aluvial or riparian landscapes.
The TLDR is that a huge low-lying basin of fertile land around what is now the British Isles was inundated at the end of the last interglacial. (What is now the North Sea/English Channel was one of those river valleys you alude to.) The British Isles today would have been the relatively cold, arid, mountainous uplands survivors migrated into when their fertile, low-lying territory got flooded out. Assuming population density in the chilly uplands (glaciers, remember!) was low, then you'd expect the frequency of surviving stone tools to be lower, reflecting that. People don't leave stuff lying around in places where people don't live!
There are plenty of very fertile, habitable areas in the same region that would have been just as attractive. So settlements? Sure. But surely a civilization is more than just a settlement?
Edit: https://commons.m.wikimedia.org/wiki/File:Post-Glacial_Sea_L...
This image shows the fastest sea level rose was around 15K years ago, which is pretty close to the Neolithic era. At it's fastest rate sea level was rising at 30 meters per 1000 years, or about 3 centimeters a year. Ok, that's a very slow moving train. It's pretty easy to get off the tracks before it hits you. Doubtful to have cause civilization decline especially when it lines up closer to the onset of agriculture.
Say you built your town on a 3% grade next to the water. That's 3 meters of descent for every 100 meters of distance. If sea level rises 50 meters, your town's ruins aren't just ~45 meters underwater, there's also almost a kilometer and a half out to sea. Nobody is looking there.
You're viewing their arguments as "Water overtook their cities and they died because of it".
Their argument is that "They died out and then water overtook their cities".
Having a civilization and keeping civilization are two different things. Once humans are capable of thinking of it, it would happen over and over again. But when it fails, the survivors might have a sort of cultural PTSD comparable to the Sodom and Gomorrah story in the Bible. People might resist trying again for generations (also, how many places could you found a civilization in a world without sanitation, metallurgy or medicine?)
That's a great question to ask of Venice, for example.
As an example: the initial human settlement of the Americas likely followed sea routes along the North, Central, and South American coasts. The region near Santa Barbara and Los Angeles, with the Channel Islands, extended much further out to sea, with some of the islands being connected to land, others being only a short distance (rather than 20-30 miles) offshore.
Settlements along both the coast and shore -- the most viable parts of these regions -- would now be utterly flooded. There are few or no traces of the initial migrations, save for a few inland settlements and venturing / foraging parties, though later settlements may exist.
Much construction and many artefacts would be of earth, wood, hides, and plant fibres, most rotting or melting away under water. There are simply few surviving remains, and those which might exist are hard to find and access.
The climate change itself would have occurred over centuries, with multiple moves and resettlements, or disappearance of entire cities. It's rare for even preindustrial civilisations to endure at their peak for more than a few centuries.
There are, of course, numerous other possible events, a synthesis, or even pure fabrication, as the basis.