Additionally, I can with very high confidence run TUI software inside an ssh session to one of my headless machines, and then further within gnu screen or tmux detach and later resume a session with everything I was doing. Even if I switch from my laptop to my desktop, or want to leave a bunch of stuff waiting for a week and come back to it later.
Once you get the key bindings for screen or tmux thoroughly memorized it's very easy.
Okay but TUIs don't matter here though right? You can easily run GUIs over the web with any box anyway if you wanted to. So how is your headless sever running 24x7 (which is what allows feature you talk about, not tmux or tuis) not the reason you can do that but TUIs are?
I have a windows box that never shuts down and i remote into it and use it GUI mode to do everything you said but with a proper GUI interface. I also use the terminal but your feature (constantly up + resume + multi device) does not need TUIs to even work.
Exactly this. Being able to attach to a tmux session over SSH and pick up where I left off is a killer feature for me. No GUI remote solution comes close in simplicity and responsiveness.