It’d be nice if you could just outbound ssh from anywhere that has a browser, but sadly this isn’t the case.
This is something I've thought about time and again, but the security issues always seem to outweigh any potential benefit from something like the OP or similar (in my opinion).
Anything more sophisticated will inspect the traffic and drop SSH connections no matter the port.
I guess if that work depends on how the proxy is set up.
If not, then this seems like an exfiltration risk your security people should fix.
Edit: Though these MITM vendors will soon have fun with DNS over https and encrypted SNI. Guess they will have to resort to being browser plugins?