How exactly could a site collect more of your data through 2fa?

Well, if you use a password manager such as bitwarden you can store your 2FA one ctrl-v away. Even if this is a less secure setup, that still prevents someone eavesdropping on your password from reusing it.

Unless you clear cookies constantly, you need to login just once in a while, where is this huge headache? Password get stolen, 2FA protect you from that.

Where does this even come from, passwords are increasingly insecure and adding another factor, especially authenticator codes, doesn't even require you to give up a single new piece of personal information. The entire thing is just adding a local code that your program of choice remembers and uses to generate the one-time password. No data collection, no proprietary software. Other areas might be doing bad shit for all I know, but this change is entirely a forced security measure because people are too bad at passwords.

After seing the frequent attempted logins on my Microsoft account, I'm "just" a lucky guess away from losing it if I do not have another thing blocking access.