I don't understand why they wouldn't deal with adult websites? Why turn business away? Doesn't make sense. I think Paypal do this as well. Couldn't websites just accept bitcoin directly and use their own bots to cash out the coins to an exchange automatically?
They are forced to say no, due to various reasons which are beyond their control. For example, the banks, which are processing the fiat payments for Bitpay will freeze their savings / current accounts, if they deal with porn websites (remember that the same had happened to merchants dealing with legal recreational marijuana in the state of Colorado).