I use email aliases to sign up to services for this reason exactly. There is no way to link the aliases, and so tracking is limited to the registered site only. The Log In with Apple solves this more elegantly

Sign In with Apple is not a panacea. UPS on-boarded me a couple of weeks after my initial registration, and the two services I’ve used it for (UPS and Etsy so far) have received my name from the initial registration process (a radio button selection and a bioauth).

I wouldn’t treat it as a privacy feature, I would treat it as a convenience feature. When I needed some extra mask straps, I installed Etsy, didn’t previously have an account. So I used Sign In with Apple to register, picked out some mask straps, and Apple Pay to pay. The whole process only took as long as it took for me to make a selection and two biometric authentications, but some of my information had to be exchanged in order to make that transaction possible.

Also I can login to UPS and Etsy without a password.