Would it be fair to say then that classical mathematics does not require computability, so it requires a lot more bookkeeping, while intuitionistic logic requires constructivism, so it's the air you live and breathe in, which is much more natural?

Intuitionistic logic is not really constrained to talking about constructive things: you just stuff everything else in the negative fragment. Does that ultimately make sense? Maybe, maybe not. Perhaps that goes too far in obscuring the inherent duality of classical logic, which is still very useful.