Your intuition is probably correct in one sense-- Holocene climatic stability was a necessary condition for the adoption of agriculture and all the other fancy cultural innovations that came along with it (big cities, complex government, craft specialization, science, etc.). Hunter-gatherers definitely understand how plants work and sometimes actively manage important plant species (e.g., sowing seeds or pruning trees), but it takes a long time for all of the technology, cultural innovations, and genetic changes in domesticates associated with full-fledged agriculture to develop. So even if there were time periods during the Pleistocene that were favorable for agriculture, they were likely too short for full-fledged ag to get off the ground. We don't see full-on ag until well into the Holocene, about 10,000 years ago.

But modern humans (Homo sapiens) replace Neandertals by about 30,000 years ago. So the development of agriculture couldn't have had anything to do with it. Even still, human hunter-gatherers out-competed Neandertal hunter-gatherers (in an ecological sense, there was probably never direct fighting), and more complex cultural innovations probably had a lot to do with it. From roughly 250,000-50,000 years ago, humans (who evolved in Africa) and Neandertals (hanging out in Europe and Asia) made essentially the same technology. But by 50,000 years ago, we see an explosion of new technologies associated with Homo sapiens. Things like art and complex symbolism (e.g., cave paintings, beads, and musical instruments), bone tools (e.g., fish hooks, harpoons, and needles), and true projectile technology that was used to hunt a wider variety of game. Modern humans carried this technology out of Africa with them and replaced Neandertals (and probably other Archaic species as well). We never see comparable technologies developed by Neandertals. Nor do they seem to adopt them after contact with modern humans.

Now, does this mean Neandertals were less intelligent than modern humans? The short answer is maybe. We occasionally see cool behavioral innovations and technology show up in Neandertal sites-- things like the stalagmite circles in the article OP posted, use of pigments, and a few bone beads and pendants. But these things are usually only found at one or a couple of sites for very brief periods of time, and never seem to spread. So even though Neandertals did innovate, those innovations were less "sticky". That could be due to a bunch of reasons. They lived at really low population densities, so small group sizes and a lack of regular contact between groups may have played some role. There could have also been differences in cognition or learning. We're not really sure. Neandertals were very successful from at least 250,000-30,000 years ago. They had a basic toolkit they used very flexibly to deal with changes in Pleistocene climates and resources. Think of it as the "Swiss Army" approach to technology.

So while raw intelligence is almost impossible to measure archaeologically, we do know that Homo sapiens rapidly developed new technologies around 50,000 years ago, and used them to out-compete Neandertals. Tools were more advanced, were more functionally specific, and when innovations happened they spread rapidly. Certainly a fundamentally different approach to adaptation, if nothing else.