Dense gases can and do sink but two factors prevent that in the atmosphere
Carbon dioxide an other dense gases and vapours do sink. This is the cause of many industrial accidents where vessels fill with some inert gas (carbon dioxide or methylene chloride vapour have killed people entering closed vessels without care and attention, for example and volcanic lakes suddenly emitting carbon dioxide have cause major natural disasters). But this does not happen in the open atmosphere to any large extent.
There are two reasons why the atmosphere is well mixed: diffusion and turbulence.
Ideal gases (and, under normal circumstances, ideality is a good approximation for atmospheric constituents) gases are entirely miscible. One molecule doesn't know what other molecules are doing and there is no strong mechanism to separate them. Diffusion alone will, ultimately, mix the gaseous components in a vessel together thoroughly. But diffusion is slow which is why natural disasters like Lake Nyos can happen and industrial accidents can happen in closed vessels. On very large scales diffusion should compete with gravity to give a concentration gradient but this is not observed in the lower atmosphere and would never affect human-scale experiments.
The lower atmosphere sees another factor that mixes gases faster: weather. Turbulent mixing operates much faster than diffusion and is very obvious on any windy day. Turbulent mixing dominates the lower atmosphere to the extent that it is a major topic of human conversation in some countries. The atmosphere is like a very large vessel that is strongly stirred, thoroughly mixing its components. This mixing force is far stronger than gravity.
If diffusion and gravity were the only factors, we would see a concentration gradient on large scales with denser gases being less common on top of tall mountains. But we don't see that effect in the lower atmosphere because turbulence is far more important. The air composition at the top of Everest is the same as at sea level (apart from being a lot less dense).
So the intuition that dense gases should separate due to gravity is correct but that effect is dominated by diffusion and turbulence. On a planetary scale there is some separation due to gravity but this effect is small and not notable even on top of the tallest mountain. On a human scale you can beat both effects with careful experiments (like filling a vessel with sulfur hexafluoride). But only under conditions where the air is still and, even then, diffusion will eventually mix the heavy gas with the air in the room, though slowly.