Author suggests using them. Resistance to Hyperreals seems to come from era before there were rigorous definitions for them.
Again, for me, this seems a little costly. If cookbook calculus is a struggle, the student probably won't take real analysis. And once you get to analysis, the fact that the reals are Archimidean grounds the whole endeavor on a very intuitive basis. By contrast, the hyperreals are not Archimedian so we can find omega bigger than every natural number. I don't personally feel like that helps intuition and indeed requires development of non-standard analysis but I admit it is pretty cool that Leibniz-style computations can be salvaged this way.
I see losing Archimedean property as more of a win than loss. It's a one simple thing that turns calculus operations into single algebraic evaluation, where you can avoid dynamic limits. Things become simpler and easier to prove.