i.e., put a cookie on the site, and trust the cookie instead of the cert for all pages except the login one? That leaves you with all the usual problems of cookie-based logins that certs are supposed to solve (revocation, risk of being stolen, etc.), and also if you want to do that you can just put a private key in localStorage and use WebCrypto on the login page. Or with username + no password + FIDO security key as your only factor, if you want a hardware token, or whatever. There's no advantage in using SSL for this.
The point of the hardware token is that the credential never touches the PC. Putting a private key in local storage is no better than a user name and password because it leaves it open to theft. The most common problem that an everyday user faces is either a data breach on the service provider end or a virus / keylogger on their PC. Hardware tokens with certs solve both those problems in terms of account access.
