Certainly the victors write the history, generally. But when the victory is so complete that there is no further threat, the victors sometimes feel they can beneficently tolerate "docile" dissent. :)
Srsly, folks: having been on various sides of such questions, at least as an interested amateur, and having wanted new-and-wacky ideas to work, and having wanted a successful return to the intuition of some of Euler's arguments ... I'd have to say that at this moment the Schwartz-Grothendieck-Bochner-Sobolev-Hilbert-Schmidt-BeppoLevi (apologies to all those I left out...) enhancement of intuitive analysis is mostly far more cost-effective than various versions of "non-standard analysis".
In brief, the ultraproduct construction and "the rules", in A. Robinson's form, are a bit tricky (for people who have external motivation... maybe lack training in model theory or set theory or...) Fat books. Even the dubious "construction of the reals" after Dedekind or Cauchy is/are less burdensome, as Rube-Goldberg as they may seem.
Nelson's "Internal Set Theory" version, as illustrated very compellingly by Alain Robert in a little book on it, as well, achieves a remarkable simplification and increased utility, in my opinion. By now, having spent some decades learning modern analysis, I do hopefully look for advantages in non-standard ideas that are not available even in the best "standard" analysis, but I cannot vouch for any ... yet.
Of course, presumably much of the "bias" is that relatively few people have been working on analysis from a non-standard viewpoint, while many-many have from a "standard" viewpoint, so the relative skewing of demonstrated advantage is not necessarily indicative...
There was a 1986 article by C. Henson and J. Keisler "on the strength of non-standard analysis", in J. Symbolic Logic, 1986, maybe cited by A. Robert?... which follows up on the idea that a well-packaged (as in Nelson) version of the set-theoretic subtley of existence of an ultraproduct is (maybe not so-) subtly stronger than the usual set-theoretic riffs we use in "doing analysis", even with AxCh as usually invoked, ... which is mostly not very serious for any specific case. I have not personally investigated this situation... but...
Again, "winning" is certainly not a reliable sign of absolute virtue. Could be a PR triumph, luck, etc. In certain arenas "winning" would be a stigma...
And certainly the excesses of the "analysis is measure theory" juggernaut are unfortunate... For that matter, a more radical opinion would be that Cantor would have found no need to invent set theory and discover problems if he'd not had a "construction of the reals".
Bottom line for me, just as one vote, one anecdotal data point: I am entirely open to non-standard methods, if they can prove themselves more effective than "standard". Yes, I've invested considerable effort to learn "standard", which, indeed, are very often badly represented in the literature, as monuments-in-the-desert to long-dead kings rather than useful viewpoints, but, nevertheless, afford some reincarnation of Euler's ideas ... albeit in different language.
That is, as a willing-to-be-an-iconoclast student of many threads, I think that (noting the bias of number-of-people working to promote and prove the utility of various viewpoints!!!) a suitably modernized (= BeppoLevi, Sobolev, Friedrichs, Schwartz, Grothendieck, et al) epsilon-delta (=classical) viewpoint can accommodate Euler's intuition adequately. So far, although Nelson's IST is much better than alternatives, I've not (yet?) seen that viewpoint produce something that was not comparably visible from the "standard" "modern" viewpoint.