News? Economist and NY Times. Recipes? Cooks Illustrated.
Of course the Web is still useful to buy things from a merchant; there, the site obviously has no incentive to be annoying. And sometimes for free stuff it's better to go to the site of someone who is already selling you something. You mention Bon Appetit and Epicurious. If you want free recipes, go to food manufacturer's website or to a grocery store's website. They are loaded with free recipes and there are no annoying ads...of course, the whole site is an ad. But it has what you need.
I understand if sites have a business model that needs annoying ads. I just don't visit them. It seems hypocritical to me to visit the site while not respecting its business model, especially when there are other sites out there whose business model is more aligned with your needs. I would rather support them with my money or clicks rather than visiting annoying places and then installing more software to suppress annoyances.
It selectively blocks add based on if they are tracking you, but I would imagine that most annoying adds are tracking you. You can also individually re-enable scripts/cookies on a third party domain per site basis.
This is what ABP attempted to do, and the reason why you see people advocating forks or alternative adblockers these days.
I really wish ad-blockers would default to an "block annoying ads only" mode; or some kind of whitelist mode; or even some way to crowdsource so that only annoying ads get blocked.