This is an interesting question! You're giving me a chance to reflect a little more than I did when I wrote that last comment.
I can only speak for myself, but it's not that I care a lot about me personally being the first one to discover some new piece of mathematics. (If I did, I'd probably still be doing research, which I'm not.) There is something very satisfying about solving a problem for yourself rather than being handed the answer, though, even if it's not an original problem. It's the same reason some people like doing sudokus, and why those people wouldn't respond well to being told that they could save a lot of time if they just used a sudoku solver or looked up the answer in the back of the book.
But that's not really what I'm getting at in the sentence you're quoting --- people are still free to solve sudokus even though sudoku solvers exist, and the same would presumably be true of proving theorems in the world we're considering. The thing I'd be most worried about is the destruction of the community of mathematicians. If math were just a fun but useless hobby, like, I don't know, whittling or something, I think there would be way fewer people doing it. And there would be even fewer people doing it as deeply and intensely as they are now when it's their full-time job. And as someone who likes math a lot, I don't love the idea of that happening.