By inserting themselves as moral arbiters
While this is effectively what is happening, and I agree with everything you said, I would like to add the primary reason why I've always heard that payment processors don't want to deal with adult content.The primary reason is because adult content has a very high percentage of disputed charges.
Typically, it's because some person's partner notices some kind of porn on the credit card statement, and the purchaser claims they were "hacked" or something and then disputes the charge. This doesn't necessarily happen a large percentage of the time, but going from e.g. 0.1% disputes (or whatever the industry norm is) to 0.2% really torpedos profit margins.
There is also some skittishness about local laws regarding morality. Credit card payments cross a lot of boundaries and various localities have wildly differing laws about adult content and so the payment processors simply don't want to risk it.
I guess what I'm saying is: the payment processors seem like the symptom of a larger problem, not the root cause.
Source: I've never worked in payment processing, but I used to run an online business with spicy content, and had to navigate this to an extent.