I can't speak for all web sites, but I reckon a combination of factors could explain why such a solution hasn't been deployed:
1. Advertising is ubiquitous, easy to integrate, and provides a safe revenue stream.
2. There is little to no infrastructure for the PPV model. Whoever builds it would need to maintain their own version of it.
3. People expect the web to be "free". This is even true within technical crowds who understand that it's really not free. And a large part of that group doesn't mind advertising.
So, really, it would require a substantial amount of effort to implement, it would add additional friction to users, and ultimately only a minority would appreciate it.
Had this model been in place from the beginning of the web, things might be different today. Alas, if my grandma had wheels...
The PPV model, like Ads, works well for websites that you're not well associated with. Random blogs and websites that you otherwise wouldn't be willing to share your credit card info with.
A critical mass of publishers would need to team up and form a cooperative/etc where a user could register once, deposit some money, and then that money would be spent every time they view an article. But that requires cooperation between competitors, which is already hard enough, and the cancer that is the advertising industry wouldn't like this potential existential threat and would be more than happy to pour fuel onto the fire to ensure it never succeeds.
What's surprising is why the card networks themselves don't get in on it. They could do so in a completely backwards-compatible manner, introducing a new card number range that only works with transactions under a certain amount and have different fraud protection/chargeback rules.
i don't believe NYT has ever tried a pay-per-view model.