I didn't want to spend too much time poking around, but it seemed odd to me too (wrt the browser-console). Cheers for the stunnel/OpenVPN thoughts, they ought to get through. It would be great if SSH could itself emulate SSL, in the modern context of increased security requirements and censorship.
In any case, you can try proxying SSH over SSL using stunnel: http://askubuntu.com/questions/423727/ssh-tunneling-over-ssl
Or you could try setting up OpenVPN, it's easy enough.