You can get most of what you want today without a protocol change. mosh only uses ssh for the initial hop, and it just runs whatever ssh is on your PATH, so your ~/.ssh/config applies. Put a ControlMaster/ControlPersist entry in there and only the first tab actually authenticates. Every tab after that reuses the socket, and the "ssh host mosh-server new" step is quick. Each tab still ends up with its own UDP connection, but that's one tiny server process and one key per tab, not a new login.

Sharing one SSP connection across several terminals is a big protocol change, and I'm deliberately keeping mosh-rs wire-compatible with upstream, so I don't want to go off on my own there. For "lots of tabs, one connection" the thing that works well is one mosh session into tmux, and tmux control mode (tmux -CC) if your terminal supports it. That gives you native tabs over a single mosh link, which is about as close to ControlMaster as you can get with mosh's design.