According to Wikipedia this particular theorem was proved almost immediately (but not published) by Dedekind, and then 10 years later proved by a 19 year old student in Cantor’s seminar. It’s not clear whether more than a handful of people cared or worked on it in between.
As for Cantor being a leading mathematician of the day: most of the other mathematicians of his time regarded this whole mess to not be mathematics at all. Poincaré called Cantor’s set theory a “disease”. It wasn’t until several decades later that a broader group of mathematicians decided that roping in set theory allowed them to conveniently hand-wave away (“oh, the set theorists will take care of that part”) a lot of thorny questions, letting them get on with their work as they had before, but now with an deflective answer whenever anyone asked such questions. ;-)
Could there exist a language where these sorts of proofs become easier to write, because that language captures the problem in a better way?
What I found interesting was that this originally stated, but not proved, by Cantor. Then a few semi-flawed attempts at proofs were made (flawed in the sense that they relied on other axioms like choice, or that they had errors).
These are leading mathematicians, including Cantor, who basically invented Set Theory from scratch. And they went through many years of struggle to arrive at a proof of a result. Which you now learn in an introductory class.
It really put into perspective for me just how hard some of these things are, and made me feel less bad when I struggle with some mathematics.