Many sites have "magic links" (they sent you a link to login via email instead of having to write a in password), but there's almost always a way to say you want to log in with your password. Sometimes, especially for touchier things, there's MFA.
> Aside from the privacy issue of forcing people to attach an email or phone number or third-party auth provider to their every account
How do you login without an email, phone number or delegating to a third party? You perform a secret magic dance? Especially for something such as booking.com which more likely than not has your bank details saved, and can wreak havoc (cancel your bookings), I'm really not sure what you want them to do.