In my country we have a saying: la matemática entra por la mano, no por los ojos -- mathematics is absorbed with your hands, not with your eyes.

I agree. Some things only stick by repeated application. This becomes more relevant the one older gets.

Depending on your level of math ability you can often learn a lot by reading textbooks and proofs very carefully and closely (my opinion)

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.

Idk, I tried to self study the first two chapters of a very hated textbook (Principles of Mathematical Analysis). If you’d have asked me if I understood the material I would have said no. But by the time I took the actual class on that material I made high As on all the exams and could call out mistakes my professor made on metric sets with counter examples.

You can't hardly learn anything by just reading or watching it.