I would say the reason to assume infinite compute was less about which parts of the problem are hard, and more about which parts can be solved without a solution to the rest.
Good solutions often have even better solutions nearby. In particular, we would expect most efficient and comprehensible finite algorithms to tend towards some nice infinite behaviour in the limit. If we find an infinite algorithm, that’s a good point to start looking for finite approximations. It is also often easier to search for an infinite algorithm than a good approximation. Backpropigation in gradient descent is a trickier algorithm than brute force search. Logical induction is more complicated to understand than brute force proof search.