I tried your solution using one remote server that I had trouble with before and it works! That's really great. This probably solves enough problems with kitty for me that I can use it until I find a better solution. (I am using Alacritty at the moment but it has other problems so that I have to use kitty as fall back for some situations.)
You likely need all of them ;-)
IIRC it "fixes" terminfo stuff "within tmux" for kitty regardless of what $TERM is set. Mind you, I was using the latest tmux at that time. tmux 3.2?
Here's the issue I had: https://github.com/kovidgoyal/kitty/issues/3018
Looks like it should be fixed on a very recent tmux; otherwise this MIGHT help:
... or not.