But despite of that, there is no reports of miners dying because on the top of the cavity the air was only nitrogen or oxygen at the bottom.
Yes and no, miners die from other causes, like lack of oxygen, sulphur poisoning, etc.
But for example you can already get a case of sulphur poisoning near Fumaroles in volcanic regions, simply because the air that has the sulphur dissolved in it is heavier than the surrounding air.
So that's the effect you have been asking for, but you still need a severe lack of wind and the right topography to make the stratification happen.
So now let's zoom out a bit. Earth has a surface. This provides friction and slows down surface winds considerably, creating the Planetary boundary layer (PBL). You can think of the PBL as 'what the free-streaming atmosphere would want to do, but then it encounters a surface'.
Only because of the friction of the surface, we have only moderately strong winds on the surface that allow for special phenomena like little valleys filled with sulphur.
Higher up in the atmosphere velocities are much faster. Thus, Reynolds numbers are higher, destabilization of flows comes easier and turbulence sets in, mixing everything.
Only at the Karman-line around 100km height, things calm down sufficiently (in terms of frictional momentum density transfer from layer to layer) that molecules start separating by weight.
So I would say your initial intuition was right, it's just less intuitive how the planetary surface plays into this.
In the oceans btw. this phenomenon plays a huge role in determining global circulation. The different salinity of water layers leads to different densities and layered flows. And because the density of water is a factor of $10^6$ higher than that of air, one would need a much stronger source of momentum to mix those layers. But such a source doesn't exist, so we get layered flows of different densities.