The key insight, which maybe people think is too obvious to say out loud, but which does need to be said out loud, is that if two quantities are the same, then they remain the same if you do the same thing to both of them. For example, if x = y, then also x - 3 = y - 3. So the entire game of algebra is to choose things to do identically to both sides of an equation. Anything about "moving" symbols is a shorthand for this process. For example, if you know "x = y + 3", then, yes, you do also know that "x - 3 = y", and someone may speak of "moving the three", but fundamentally what you're actually doing is not "moving" anything, but rather subtracting three from both the quantity on the left, and from the quantity on the right. You picked an operation to perform on both quantities, to get another true statement.

The exercise that finally made this make sense, was to painstakingly walk through simple proofs, in tiny tiny steps, starting from small numbers of self-evident postulates. This is in the tradition of Euclid's Elements, which has a totally different vibe from most math education. It was important to do this so slowly and pedantically that you would be embarrassed to do it -- you would worry people would think you are stupid -- if the correct norms had not been established. You need to establish the norm that leaping ahead and skipping steps is not a sign of intelligence but is instead sloppiness, ugliness, something missing, a flaw in the argument you're constructing.