Does clay absorb water because of polarity?
I am no expert regarding clay (which is a surprisingly varied field) and will therefore give a more general answer. I suspect that there is also quite complex interaction with ions on the surfaces of the layers and such things. 

Materials can be generally categorized on a rather continuous scale, the extremes of which are called hydrophilic and hydrophobic (or lipophobic/lipophilic which are synonyms). 

If something is hydrophilic, it "likes" to be wet, i.e. it is energetically favourable for the surface to be wet because this minimized surface energy of the system. The opposite applies to hydrophobic substances. 

As you have observed yourself, clay is quite hydrophilic. 

One can indeed estimate whether a material is hydrophilic based on its chemistry: water likes to form hydrogen bonds and water complexes (e.g. around Na+). Therefore things are more hydrophilic the more of said things they have. A good example is that sugars and carbohydrates are very soluble in water despite being large, whereas even small alkanes are barely soluble. 

The explanation involving air pressure makes no sense, because to move matter there has to be a pressure difference. But if you imagine clay as a porous substance, the pores might as well be filled with air and there is an equilibrium. In fact (as easily observed when burning earthware) clay gets more voluminous with higher water content.