But some sites that you would want to use this service for explicitly block it (e.g, Bloomberg seems to do this). There's also a small number of other sites that I've had problems with.
Sometimes the customer support is mediocre.
Does this still happen with the new kind of cards Privacy.com recently started issuing? https://support.privacy.com/hc/en-us/categories/441448782555...
It makes sense that merchants could refuse prepaid cards, but now that these are regular charge cards, I'd be surprised if the merchant agreement with the payment networks let them refuse cards arbitrarily.