Yes, clearly, a bit after I asked, I learned how to use intuition, and at some point, it became rote. But the bigger point is that this is a special case -- in logic, and in math, there are a lot of truth-preserving transformations, and choosing a sequence of transformations is what doing math is. That interesting interface between logic-as-rigid and math-as-something-exploratory is a big part of the fun in math, and what led me to do enough math that led to a published paper. Of course, after that, I went into software engineering, but I never forgot that initial sensation of "oh my god that is awesome" the first time Moshe_1992 learned that there is no such thing as "moving the 1 from one side of the equation to the other" except as a high-level abstraction.

I can't see how one could get through even basic algebra without developing this intuition, though o.o

And yet so many do. Numbers are horribly scarce in this area, though. Sometimes I get desperate.

Anecdotally, I do remember saying something very similar to high school peers, in a manner that assumed they already understood it, and seeing their face contort in exactly the same manner that it would have had I suddenly metamorphosed into a polar bear and started writing all the equations for fourier and laplace transforms using trails of volcanic ash in mid-air.

This was years after basic algebra had already been taught according to the curriculum, and we were now beginning pre-calculus (i.e. last year of secondary / high school here in québec, calculus itself is never touched in high school level courses).