Mastery comes from problem solving and practice - not from reading books. So, I would advise students to limit what you read, but spend a lot of time in problem solving. Get the basics in place. Start with euclid's elements and master that first.
I think Euclid is fine as a historical document and interesting in a broad sense, but its kind of silly to start with a document from 300 BCE when you could start with, for example, "How to Solve It" (Polya). And even that text could use a rewrite to make it much more readable.
I have completed substantial education in both mathematics and physics and I would say one of the weaknesses of the standard system of courses in physics is that it more or less recapitulates the development of physics in a historical shape, which substantially obscures mathematical structures which are shared between disciplines developed at different times. For example, unless you were lucky or very curious you might never appreciate that the bras and covectors relate to kets and vectors in fundamentally the same way. You might only have had a vague sense that physics involves making a lot of sandwiches.
Don't get me wrong, I'd be delighted if my kid's school broke out The Elements, but I just don't think its an obvious pedagogical strategy to start math instruction (self or otherwise) there.
You make a very good point. The historical trajectory of a field's development is an extremely haphazard presentation of the ideas. It belongs in a separate curriculum, and I'm grateful that the history of science exists as field for that purpose. It's nice to study the history after one has understood the material; that way we can see what ideas prospered and faltered, and what might be ready for reinterrogation.
Why should we study antiquated and easily falsifiable models before we get to our modern and less-easily falsifiable models?
Drawing on common intro chemistry: the plum-pudding model of the atom is cute in historical context, but a real distraction from what our best understanding of what atoms are, for which we have much better evidence than the helium-nucleus scattering experiment that first suggested a dense, charged nucleus in gold atoms. We really only need the old plum pudding as a counter example, yet fail to explain the experiment in enough detail to justify including it. What probably began as a fastidious attempt to provide full context to a landmark experiment has at this point completely degenerated into historical trivia about the structure of British desserts, and yet remains prominent in educational material.
Math is a bit better off in this respect.