Compromise is always the game.
You either comprise by not using YouTube anymore, or you compromise by giving them the most tightly restricted information they’ll accept.

You’ll never have perfect privacy. I guarentee Google already has a supressing amount of info on you. At least this is officially protected. Once you give the card as proof, you can probably delete it from their saved data. And they have to honor that, both by law and contracts with Visa etc.