I really wanted to be a customer, but when I visited privacy.com and saw the number of 3P services they load up (thanks uBlock Origin) I realized that they talk the talk but don't walk the walk.
I use the service. The name of the service is sort of misleading advertising. In reality what you get is protection from theft of a card number when shopping online (this is what happened to make me decide to use the service), spend limits on reusable cards (for something you pay a recurring payment for, this comes in handy when a merchant raises their rates and doesn't tell you) and you can use any name and region and so protect your identity from being sold by merchants to advertising firms.
Your bank still has access to everything (and they sell everything, of course) and the service itself also has access to everything. But all in all it is a good layer of insulation from merchants and vendors and has some useful features. Think of it like a VPN service for card payments.