You lack some imagination. One could verify their identity against a government service once every let’s say 5 years. You get a public link/code that you can share with employers (or banks or real state agencies or…) so that they can use the link/code against a government service to verify that you are who you claim to be.
Adding a session token in the middle of the process of a website getting your government ID won't stop the website getting your government ID though? Either way the website learns your full name and address, and you can only ever have one account - which will make you extra careful not to violate any real or imagined rules - the chilling effect on speech is still there.
