It depends on your definition of learning. If by learning you mean remembering how one author defined and proved various things then sure. If by learning you mean gaining the ability to do what the author did except for new problems, then at some point you will have to grapple with the fact that coming up with new ideas is a very different mental exercise from recalling things you've read.

A supporting factor in the necessity of exercises for learning to actually do the math is that authors often hide important technical tricks in the exercises.

Of course, a sufficiently intelligent person could learn anything by reading tea leaves and then deriving on their own whatever it was they wanted to learn, so in a sense you can learn math by doing anything.

I disagree. Generally speaking students aren't able to teach themselves mathematics on their own until they hit the graduate level. That's not a hard rule, it just depends on mathematical maturity.

But everyone I know who is able to teach themselves math without lectures or a professor still does the exercises in the textbook. Math isn't a spectator sport, it's really important to do active learning.

This depends on the textbooks. It is very, very hard to find a textbook that is (a) well written, (b) suitable for self-study and (c) affordable.