Sometimes with math the proof you read in the textbook is not the first proof someone thought of. It may not be the clearest. It may be the shortest. This is not a new thought, but I can't remotely remember where I read/heard it. The interesting bit is that someone created that second proof by a second interaction with the problem, and aiming for something slightly different.
I like to picture this as speedrunning. Once you've got the first proof, you know more about what the "routes" in the "game" might be and you know the punchline, the ending. So now you're exploring the space in between, trying to find the shortest path. (Ideally, also the clearest path, but those are rarely exactly the same with math.)