My understanding of the history is that the developers of tmux were using screen to develop tmux, because tmux didn't exist yet, so they needed a non-conflicting keystroke.

Obviously they could have switched before a general release, but I guess they had gotten used to it.

1. Outer tmux on local/primary machine, for your usual daily use-case (i.e. managing multiple terminal sessions).

2. In outer tmux, ssh to some other machine you need to administrate.

3. screen because you need multiple terminal sessions on that machine, or need detachability/reconnectability for some long-running process.

Also, many older production machines don't have tmux, so nesting a screen session happens. This is starting to not be a thing, but it was a thing for the trailing 10 years.

At work, my desktop has tmux, but the servers we deploy on only have screen.

Hitting C-a twice sends it to the terminal.

Close... "C-a a" sends a real "C-a", while "C-a C-a" swaps with your most recent window (it's fast and convenient to type).