Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GLn over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic p, a commutative group of order prime to p, and a p-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

My own speculation involves Kolmogorov complexity, in general agreement of your position. Basically, when things are controlled by small numbers (dimension, size, whatever), there often isn’t enough “room” for generic phenomena to happen. Because of the lack of room, unlikely events are forced. Which turns out to be consistent with Corfield and Gelfand. In other words, I think both explanations can be made consistent with each other.