I wouldn't want to rigorously teach Calculus using *R unless the relevant proofs are reasonably accessible (which, given that we're talking Number Theory, seems like a remote possibility), but it's wonderful how much simpler they make the arguments if you work with them.
The only thing that isn't easily accessible is the proof of transference, namely that what's true for star-R (don't know how to make asterisks format nicely) is also true for R.
Our use of R rather than star-R is merely a quirk of history - we managed to justify R to ourselves first, hence it's the "standard".
(If you look at the other axioms, the hardest one to justify is the bounding axiom, and its necessity is reinforced by proof. Every other axiom fits in with the student's understanding of arithmetic.)