The alternative to advertising is paid subscription content. I don’t see how you’re going to achieve greater privacy by giving someone your credit card number.
Civilized Internet economies have moved way past people giving every service or merchant their credit card numbers. There is a number of alternative payment schemes which share less information with the money recipient, and doesn't let them pull money off one's account.
Can you please explicitly list those alternative payment methods you had in mind?
When I last researched this, alternatives like micropayments, Flattr, cryptocurrency, digital wallet donations, etc are not viable revenue amounts in comparison to credit-card subscriptions or sponsored ads.
As for the mobile phone payments that are more popular than credit-cards in China and Africa, those are not privacy-preserving from what I can tell.
Well, in my country there's a number of payment intermediaries like Przelewy24, PayU and a bunch of others, which proxy a payment without themselves receiving your card number either. The way it's constructed, you get redirected to your own bank's webpage, where you log in just to confirm a pre-filled transaction from your account to the intermediary service.
Then there's also Blik, which is the same except I don't even have to log in on the web page; I read a random six-digit code off my bank's app, enter it on the payment intermediary site, confirm again on the phone, and money moves.
Why? First of all, why must there be a dichotomy here? Second, why must I give each site my credit card number directly? Why can't we do that differently?
The conventional wisdom is that ads are more convenient. I hope that, sooner or later, this will stop being true and we'll have a variety of better ways of supporting content.