I think you'll be pleasantly surprised if you look into what tmux is actually capable of. It's extremely customizable, featureful, and well-documented. All of the complaints you've raised in this thread can actually be solved with a bit of configuration.
> TMux itself contains a terminal emulator, and not everything works correctly in TMux
tmux doesn't merely contain a terminal emulator. Rather, it is a terminal emulator. All terminal emulators have slight differences in behavior and tmux is no exception. But since tmux is actively developed, I haven't personally encountered any deal breaking issues.