It's an interesting idea. However, I'm not comfortable with a third party having all that information. Some banks issue "corporate" cards, with numerous "employee" cards. I already trust the bank, after all. So what else does Privacy.com provide that's worth the risk? They're still subject to KYC, right? So there's no strong privacy. Or am I missing something?
We are still subject to US AML / KYC laws. But the cool thing about these cards is you can use any name or billing info you want with them, so don't have to worry about your info getting leaked if some website you bought an indie game / song / whatever from 6 months ago got hacked.
OK, so that would have protected Ashley Madison users. Because none of the likely interested parties (partners, private investigators, etc) would have leverage to get information from you. Same for users buying porn, unless they get investigated for child porn. That's not a KYC issue, but there will be a subpoena. And I'm assuming that you must comply with all subpoenas.
Edit: I wonder what your burden would be in bankruptcy cases.
Ashley Madison and porn users are easy targets. But broadly-speaking, we just think you just shouldn't have to share your personal info with a random merchant you want to buy something from.
It's anti-privacy in the guise of being anti-fraud.
Still, users arguably need to trust you more than they trust traditional credit card companies. But it's about the same as PayPal, I guess. They often do have access to users' bank accounts.
So anyway, I get the point. It's a useful service.