You cannot explain Polaris with normal variable thickness of the atmosphere, as far as I can tell.
As long as the amount that the atmosphere blocks light is constant based on distance, and not exponential, then the variable distances between the north star and you on the flat earth should, yes, lead to the star being less visible, and eventually not visible at all, but not entirely invisible.
I.E:

Of course, based on the Pythagorean Theorem, the distance the light would have to travel to Observer A, south of the Equator, is SQRT(SN^2+AN^2). For Observer B, North of the equator, it would be SQRT(SN^2+BN^2).
Obviously, based on the density of atmosphere theory, it would be less visible for observer A than observer B. Conceivably, this could be to the point that it is not visible at all.
Based on this theory, then, the star would loose its visibility directly at the equator. Hence, as is obvious mathematically, the visibility would get progressively worse form the North Pole as you moved towards the equator until it reached zero at the equator.
However, this is not the case: Any direct observation will tell you that the loss of visibility is not gradual, but sudden at the equator. This either means there is a very large change in the density of air at the equator (which would be extremely easy to notice

) or that this theory does not adequately explain the visibility of Polaris.
Of course, there may be other, more accurate ideas. Density of the atmosphere is not accurate or even remotley provable.