> I mean... isn't the point of academic writing to communicate research and results already performed?

Obviously mathematicians don't leave "exercises for the reader" in research papers, but it is common in textbooks or expository writing.

> I always felt that phrase has no place in the internet age, where the concept of a "page limit" is laughable and a simple hyperlink can point me to chapters upon chapters of appendices.

The constraint is not page length, of course, but time. It is a much better use of a mathematician's time to leave easily reproducible proofs to the reader, and focus on the interesting ideas.

Anyway, doing these kinds of exercises is actually very useful in understanding new ideas.