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I theorize this is a good way to build mathematical maturity, at least the "parse advanced math" part. I remember when I became mathematically mature enough to read Math Wikipedia, I want to go further in this direction till I can read math-y papers like Wikipedia.
#3 is good. another good reason is so you have enough mathematical maturity to understand fancy theoretical results.
I'm probably overestimating the importance of #4, really I just like having the ability to pick up a random undergrad/early-grad math book and understand what's going on, and I'd like to extend that further up the tree :)
3Ulisse Mini
(Note; I haven't finished any of them)
Quantum computing since Democritus is great, I understand Godel's results now! And a bunch of complexity stuff I'm still wrapping my head around.
The Road to Reality is great, I can pretend to know complex analysis after reading chapters 5,7,8 and most people can't tell the difference! Here's a solution to a problem in chapter 7 I wrote up.
I've only skimmed parts of the Princeton guides, and different articles are written by different authors—but Tao's explanation of compactness (also in the book) is fantastic, I don't remember specific other things I read.
Started reading "All the math you missed" but stopped before I got to the new parts, did review linear algebra usefully though. Will definitely read more at some point.
I read some of The Napkin's guide to Group Theory, but not much else. Got a great joke from it: