[Squithrilve]

Poor support of SSL certificates in browsers is commonly attributed to bad UX but the real reason is that they are credentials that can be re-used by multiple services to track you as an individual. Newer standards like U2F completely compartmentalizes origins so that if you register to service X a different service Y will not know who you are.

SSL certificates also doesn't work in HTTP/2 (because of multiplexing multiple requests).

Benefits include storing private key in a hardware tokens, most OSes support them out of the box. You can just plug your token into USB port, visit site that requests a client certificate, enter PIN and be done (e.g. Yubico PIV applet).

HTML also has/had <keygen> element that would generate private key in a browser, send the public key to be signed to a webpage essentially creating private/public key credentials but that is being removed from browsers.

For inter-service communication I'd definitely consider using SSL client certificates pinning private keys e.g. to TPM but regular users can't be bothered with it.

If you're interested check out Token Binding that makes tokens (cookies, etc.) bound to TLS connections essentially providing security of client certificates for tokens.