I am not a web developer, but I would disagree with that.
Either web standards respect privacy or they don't, but I would not sacrifice privacy for anything.
Firefox was right to prevent tracking, it highlights how webstandards are just not good. I something doesn't work properly in a firefox private window, to me it should not exist.
It's also a bit disheartening to see Matrix putting all that "Log in with Google", Apple, Facebook etc so prominently on their login page. The whole idea of decentralised services was getting out of those walled gardens.
I struggle to see why I should trust it with those things but not the account password.
Bubbling up these architectural details to the front end is a symptom of the webdev cargo cult coming up with broken ideas that get fossilized as the status quo.
The alternative would be something where I enter my Google username/password on random websites, and trust that they will forward it to Google and not do anything nefarious. This is less secure and less private.
That's not something companies like Matrix can use. If you're installing software already, why not skip the browser engine and install a full Matrix client instead?
Privacy Pass is currently being standardized by the IETF, so we may see more widespread adoption eventually: https://privacypass.github.io/
If a Privacy Pass token is needed for access to your email, then redeeming the token tells the service you (the client) can access your email. That's identified you.