In practice, a proof contains many steps that are highly non-obvious unless you have a deep understanding of the subject. This is done not just for the sake of

In practice, a proof contains many steps that are highly non-obvious unless you have a deep understanding of the subject. This is done not just for the sake of brevity, but also clarity. In a proof, there are a few key ideas you want to show. Around that, you need glue to convince the reader that your key ideas work, and save them from some the tedium and hard work of verifying that stuff. A 1/4 page proof, when made as simple as you suggested, could easily bloom to 2 pages. At this point, your reader can no longer find those few key ideas, and is instead lost in the rigor.