Why air do not suffer density stratification of its gaseous components?
So, naively, one would expect the heavier gasses to pool in the lower atmosphere and the light ones at the top.


That's overly naive. That same naive expectation would lead one to think that uranium should be concentrated in the Earth's core, or that a mix of water and ethanol should be differentiated with water on the bottom and ethanol on top. Uranium is instead concentrated in the Earth's crust, and water and ethanol readily mix. Entropy and chemistry can lead to counterintuitive results.

That most certainly is the case with gases. The entropically-favored distribution is a thorough mix. Attaining this thorough mix can take a long time if only molecular diffusion is in play. Turbulent mixing makes the mixing proceed much more rapidly.

There are places where this entropically-favored distribution does not occur. Gases can be concentrated in mines and in basements. For example, mines have problems with concentrations of methane and carbon monoxide (both less dense than air), but also with concentrations of carbon dioxide and sulfur sulfide (both more dense than air). Basements in areas with lots of granite have problems with radon (significantly more dense than air). Gases can concentrate in mines and basements due to a diminished turbulent mixing. This enables processes that produce gases to overwhelm the slow molecular dispersion process, resulting in an entropically-disfavors distribution.

The Earth's upper atmosphere is also differentiated. Here, the extreme rarity of the upper atmosphere means that turbulent mixing is highly attenuated. Extreme solar radiation (high UV and x-rays) produces gases by splitting molecular oxygen, water vapor, and to a lesser extent, nitrogen, into constituent parts. The very long mean free path in the upper atmosphere means that the lighter components move higher than do the heavier components.