On top of the fact that they're entirely different protocols and tools designed for entirely different purposes, browsers already support virtually everything SSH does. File transfers, authentication, client certs, multiplexing, key pinning, etc. There is no need to use SSH, and if you did, it would be slower, less secure, and generally more annoying than using the existing tools built into browsers.
I find it aggravating when users read the manual to some software and think they have discovered fire.