It's probably not payment processors as such that care, but rather countries. Payment processors are just a convenient lever. Payment processors are vulnerable to national pressure, and random commercial website is vulnerable to pressure from payment processors.
Payment processors do care, but not for the reasons people seem to assume. NSFW purchases have disproportionately high rates of buyer remorse and charge reversals, which understandably make them much less desirable for anyone to deal with.
Prudishness may also play a part, but the chargeback rate is a major factor.
I wonder how often Visa and Mastercard had to deal with those problems up to the early 2000s, with nearly every porn site being paid access and none of them accepting Paypal, as far as I remember.
The industry found another way through a company called CCBill, that essentially became the adult industry's preferred payment gateway. They took a lot of the heat off the unsavoriness of it.