I feel like your claim is begging the question.

It's certainly not true in principle that most problems in P have a small exponent. There is no shortage of graph related algorithms in P that have absolutely absurd exponents, like on the order of 10^100. It is true that practical decision problems that are in P have small exponents but that's exactly the point being made, namely that problems in P with large exponents are not practical and hence don't get much attention.