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You absolutely can do client authn in HTTPS, but it's rarely done outside intranet apps and high-security banking.

Your complaint about a login system is about its authentication, in a discussion on encryption and key exchange?

The obvious point being made is that going from SMS -> E2E should be as seamless as from telnet -> ssh.

SMS and messaging systems generally do require some level of authentication so you're making a ridiculous comparison.

auth_pam/auth_basic in the nginx config and the http/https does authentication.

And far less space to mess things up than with coding a fancy login page.