I largely agree with your points, but i'd say it's more "intuition building". To make a silly example, I don't know C# at all, in fact I have never written a single line of C#, but I'm very confident I could get up to speed relatively quickly because I know Java fairly well. This is simply due to the fact that after a while you just "get" how things work.

Similarly, even though I'm by no means a mathematician, I feel, at least to a limited extent, like I "get" math.

This kind of intuition, innate feel for how things work in a certain field of study lasts way longer than the details of a specific proof or the code for a specific algorithm, and I feel it's the actual output of higher education.