I've encountered several sites that broken with blocked third-party cookies. Most of the website for one local bank around here and a payment form for a local puppet theater are the ones that come to mind offhand.

Typically, it'll be small sites that are outsourcing part of their site to a third party but don't want to open a separate tab for that which will be affected by this, obviously. If you only browse major sites doing everything in-house you're not going to run into problems.

Some sites do break. The problem is that by the time you run into one you'll have forgotten you changed this setting.

That's what I do too. Deny all third-party cookies, plus uBlock Origin with a sizable filter list and prefetching and link auditing blocked, video autoplay disabled gets me most of the security benefits with minimal setup and cognitive load. Privacy is not much better than running vanilla because although 3rd party cookies are blocked, most sites can still identify by device fingerprinting. That can only be blocked by not running Javascript, but that's personally too much of a hassle to handle and unbreak for practically every site out there.

If you ever visit the Facebook website directly the cookie is set because then it’s a first party cookie. Then it keeps sending this cookie on other sites.