> I’ve heard this is apparently less about morality and more about payments getting contested by men getting caught by their spouses

This is a common myth. The concerted effort that we've seen over the last 5-10 years in particular is the direct consequence of intense lobbying from a handful of groups that are openly backed by or aligned with right-wing religious groups, especially (but not exclusively) evangelicals.

In case you have any doubts about whether "morality" is their motivation, one of the groups was literally called "Morality in Media", before renaming to the more official-sounding National Center on Sexual Exploitation. Despite the new name, they actually don't care very much about "sexual exploitation" as most reasonable people would define the term (such as child abuse) but instead consider all sex work to be "exploitation" and aim to ban legal sex work.