Can you elaborate further on what gain I would have by having the tmux UI "rendered locally" on my system vs the SSH host? I'm obviously missing something based on your description.

For example: SSH into server, start tmux on that system. If I am disconnected for some reason, SSH back into server, tmux attach -t 0 (or whatever ID I have) and I've recovered my session. Cut/copy/paste, mouse use, etc., are all treated as they are through any SSH session.