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.
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.
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.
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.