When reading older math texts I've seen it advised "these should be learned by heart"

This idea of learning "by heart" is often conflated with memorization. The results are similar. You can reproduce the proof perfectly without looking at any notes.

To know proof "by heart" is very different memorization. It would be possible to memorize a proof in a language you didn't know-- you can only know something by heart in the particular idiosyncrasies of your own mind. It's literally a part of you.