In general the problem of colliding keybindings is a hard one, and one we plan to address in the near future as we chip away at some technical debt that stands in our way.
Too much overlap in functionality. And windows in windows in windows in ...and another set of keybindings to memorize.
To get a persistent session, just run Emacs in screen.
(Yes, Emacs can still run in terminals if you want to. You just don’t need to keep pretending it’s 1979.)
I rarely bother splitting a tab into multiple panes (with either tool).
I like tmux's model where I can switch to a different session quickly, or switch to a pane quickly.
I've tried using vterm in Emacs, but it doesn't stick as nicely to how I think about things.
Terminal support in emacs isn't super great so there's definitely room for improvement there. Maybe there's even room for Zillij and emacs to integrate via plugins. That could be quite interesting. I still use emacs and tmux side by side without any integration.