I was going to say, but zellij doesn't connect the session to the GUI. However, I did not know about kitty's --use-foreground-process, which doesn't solve the problem completely but may be good enough.
I used screen until like 2010 or so -- but you triggered me to look back at it again today. One nicety there now is that you can easily get a dump of what's showing in the pty with screen -S <session_name> -X hardcopy. This is actually somewhat hard to do in tmux, but it's a thing you want your AI agents to do all the time. I'm curious if claude / codex are comfortable with screen, I'll check it out.