By today's standards yes, but it used to be really progressive in that it broke with the standards of teaching (proper) logic first and then building the complex numbers and concept of differentiation from naught but logical calculus. Rudin even did away with the Dedekind cuts from the regular course material and relegated them to an appendix to give an easier intro into the field of real numbers. This was heresy at a time when mathematically sound teaching had to be logically sound as well.

I think its still a good reference for exam prep or when you have to look at what exactly stokes theorem said again but I agree with your point that it is not a good introductory material into maths in general. Proof writing is the real art that needs to be the first step into math but I don't think that can be taught by book alone, you need someone to read and correct and criticise your proofs if you want.

But "Principals of mathematical analysis" could have been the title of a brochure to get people interested in some religion called "mathematism" or something so I thought it worked best for the bit.