I've been burnt one too many times by trying to "upgrade" to HTTP/2. It simply broke client certificate authentication and NTLM authentication, leaving only cookie-based authentication fully functional. Can you guess which two of the three popular authentication mechanisms Google doesn't use? It's not a protocol designed to advance the Internet, it's a protocol designed by Google to shave 1% off their network bill.
Google is hardly alone. Non-cookie-based authentication mechanisms aren't used (and aren't even usable) on any other public web site, either.
TLS client certificates have always been a UX nightmare. There is no standard UI for creating, installing, or selecting a certificate to use to authenticate to a web server; many browsers (especially on mobile devices) don't even support those features. There's no way to log out without closing the browser. There's no way for an average user to transfer a certificate from one device to another.
NTLM is simply irrelevant outside the scope of a Windows network. Other HTTP password mechanisms have many of the same failings as client certificates -- the UI is clunky and sometimes unavailable, and there's no standard way to log out.