I may also want to ctrl-z Vim to run a few quick commands before resuming Vim with fg. Using tmux panes, I'd only get a small window to run those few commands, which doesn't reflect my intent. (Yes, I could use another tmux window for this, but I like this way.)
I suspect it would also mean more keystrokes.
Amp also supports suspending (hit "z" in normal mode) as I use that workflow all the time. As for the small window issue, you can "zoom" a tmux panel with <leader>-z. I get that this is less than ideal if you're used to the suspend action collapsing all panes by default (and expanding them on resume). Workflows are tricky; to each their own, I guess! :)
By "native" does that mean an X-server's clipboard? So when you're running it on a machine that has no X-server running (a typical server), there is no clipboard?
In that scenario, I would compromise by opening both files in the same Amp process and switch between them, but that's not the same as seeing both at the same time.