> My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics.
NO, NO, NO.
There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly what Linear Algebra would look like if instead of finite dimensional vector spaces we considered infinite dimensional vector spaces. It opens the rigorous path to non-linear optimization, analysis of pdes, numerical analysis, control theory, an so on. What this article mentions is a way to work around things, but nowhere near an undergraduate degree in mathematics.
I'm astonished that the PDE section has such books, they look like the calculus aspect of partial differential equations. A more appropriate book would be L. C. Evans' Partial Differential Equations. Same with ODEs, no mention of Barreira's or Coddington & Levinson's books.
This is certainly not universally the case, even in very well-regarded departments. The University of Chicago, for example, does not require it: http://collegecatalog.uchicago.edu/thecollege/mathematics/.