Yes, sort of. There are a number of problems that basically boil down to "what's the smallest value that has property <x>", which can be solved via brute force

Yes, sort of. There are a number of problems that basically boil down to "what's the smallest value that has property <x>", which can be solved via brute force - or rather, could be, except that naive brute force would take too long, and no-one has figured out any tricks to make it fast enough to be practical. For example, calculating the value of any but the first few Ramsey numbers (the minimum number of vertices v = R(m, n) such that all undirected simple graphs of order v contain a clique of order m or an independent set of order n). There are also a lot of problems that could be proven or disproven via (counter)example - something like the Collatz conjecture, for instance.