I will never understand why SSH in such tools isn't native but always via some weird web UI...

I used to work for a company who allowed SSH only after jumping through Citrix => RDP => Putty => Jumphost => Target server.

Incredibly painful, also considering that each layer had a different keymap

I think that's because what you're really looking for isn't a jump server but a zero-trust network like cloudflare access or beyondcorp. You want authorized native connections, not proxies in the typical sense (although they do end up being proxies but more like a L3 proxy not L7)