Ujm, as far as i know, people who look for Orthodox rabbis for kiddushin will generally do things like marriage, divorce, funerals, etc…with Orthodox rabbis; it’s a mentality that “when” they do something religious, it’s authentic.
But im sure there are many such disgruntled women who ger a civil divorce and remarry r”l.
And yes, according to rav Henkin there’s a fairly large population of mamzerim, especially in America. But rav Moshe was rav moshe – he had the plaitzas to allow it, and i believe this is accepted by kiruv rabbis, but that’s not something i know for sure and i don’t expect anyone on here to take what is said on an online forum seriously enough to pasken such a shailoh.
As for the issue of staying married to someone who’s not frum – for many BTs, having to divorce would send them way backwards, they would lose control over the chinuch of their kids, and they may drop it altogether – it’s dinei nefashos to tell a BT to do such a thing. While of course it’s not ideal to stay married, if the home is kosher and the frum spouse can keep shabbos and taharas mishpacha, I don’t know of any reason good enough to tell a person to make such a sacrifice if they’re not holding by the madrega of wanting to do so themselves. And even if they want to, there’s the children to consider.