This mostly started as an extended comment, but became far too long to be one.
If you want any context as to my perspective, I recently came out of my undergrad in mathematics, and will be entering a doctoral program this coming autumn. My school isn't particularly prestigious or anything so I've probably not had the most rigorous education in math (though I've done some self-studying and took nearly every math course at my school).
I mostly say that because there's likely a lot of naivete in what follows and it likely gives off a whole "wet behind the ears" vibe to those who know better.
Every text has its merits and its drawbacks; no one textbook can work well with every type of learner, nor can a learner easily adapt to any given textbook. If this student is so confident in the superiority of Rudin's work or these other works, I have to wonder if he can cite the strengths and failings of these works -- or perhaps he is just going off their reputation? I know that's part of why I've wanted to go through some of these works that are heralded as the best in math, or at least very commonly used - Rudin's analysis texts, Euclid's Elements, Spivak's Calculus, Jacobson's Basic Algebra, etc. etc. etc.
In the case of Rudin, from what I've heard (I've yet to read it myself sadly), I think I've heard its main merits described as being concise but challenging, leaving a lot of work to the reader in terms of filling in the blanks and understanding the motivations, with a variety of sometimes difficult exercises. That if nothing else means it has a steep learning curve. When I hear that, I don't look at it as a good book for beginners and students in their first analysis class unless they really want to test themselves, or they're really bright, or they have an amazing professor.
Rather, I see that as a sort of way to test your knowledge and your ability after having grasped the basics from simpler texts. It certainly does seem a good book in that light, as a sort of "big step" for those loving analysis; the conciseness also likely means it's a good text to keep around for reference, for instance.
I feel that the student in question might have a point, however, in mathematical maturity. You can't always have everything proverbially spoon-fed to you -- or, more clearly, if you need to study some advanced, narrow subject for your field (research or whatever), it would take too much time to find an easy text, study that, and work your way up. At some point in your mathematical career, I feel there is the belief that you've grasped the overall "meta mathematics," as it were -- how to learn, how to explain, how to communicate, how to fill in the blanks, all stuff you've learned through experience, in a proper rigorous and mathematical way -- and that you at that point don't need every little detail told explicitly, that you can fill them in on your own.
I'm not sure at what point that comes for a person. It's probably different for every mathematician. Everyone is a different learner -- they learn at different speeds and in different ways. I think that's the main thing you should take away from this discussion.
Do I think it will drastically affect your mathematical career if you learned from someone else as opposed to Rudin? I doubt it.
Do I think reading and working through Rudin might be a nice way to cement your knowledge, and if nothing else help ease some of your worries on the matter? It might.