When you say login would break, what do you mean? Just that it would be less convenient, i.e. you have to login to your identity provider once for each first party? Or are you suggesting that the redirect flow for OpenID Connect or oauth wouldn't work at all?

Yet another reason why you should avoid using service to log into another as much as possible.

I'm really of two minds about this. On the one hand you can use Facebook or Google to log in and then have them know what sites you log into. Or you can use a site's native login and be pretty sure that at some point you're going to get a "we take security very seriously" notice after they screw up and expose all their user data.

If you use a password manager with different passwords for each site it doesn't matter (much). Either they steal your password (usually hashed), or your OAuth information.

The minor difference is that with OAuth they cannot log-in to the site pretending to be you, but if they don't hash the passwords, or use weak method to do it, then it's possible.

Regarding other user data stored in the DB (which is usually more valuable than just the passwords) there is no difference between the login method.

I don't really see how logging in with a different system prevents your user data from being exposed?