I agree there can be some theory crafting in the search for a counterexample, but in general I think it is easier to search for.

For proving a proposition P I have to show for all x P(x), but for contradiction I only have to show that there exists an x such that not P(x).

While I agree there could be a lot of theory crafting to reduce the search space of possible x's to find not P(x), but with for all x P(x) you have to be able to produce a larger framework that explains why no counter example exists.