>Have mathematicians become "less savvy" since the invention of the abacus or calculator?

The abacus and calculator doesn't do high order thinking or prove theorems for you. Math as a field aren't about manually adding numbers together or doing simple algebra formulas, and wheren't that even 2000 years ago.

Those calculators are to math what something like an IDE or a helper library is to programming.

The analogy would be with theorem provers, and even better, with the exact same tool as in the programming case: LLM doing the proofs.

If mathematicians had LLMs and theorem provers do their job, they'd certainly have gotten "less savvy". And if they let them write their coursework, starting from university, they'd might never even become savvy enough in the first place.