Yeah, you make some good points. I think at the end of the day both can work - anything you can do in the standard analysis setting can in principle be done in NSA too by the transfer theorem, it's just less travelled ground and maybe the requisite techniques aren't as well known.

Integration is very similar, by the way - you just do Riemann sums with a hyperfinite number of partitions of infinitesmal width. It sounds weird, and it is, but it makes it very easy to understand why integrating f(x) against a delta function gives you f(0), for instance, without having to justify it with limits or a bunch of deeper theory.