Tmux breaks interacting with the clipboard so much that it has its own dedicated Wiki page dealing with all of the different issues and settings: https://github.com/tmux/tmux/wiki/Clipboard
It literally doesn't though. The wiki page is for copying tmux copy-mode's internal selection. The selection and copying mechanisms of the terminal emulator still work just as well.
tmux/screen do not break copy-paste. And that tmux/screen have their own scrollback is a feature.
The selling point for shpool is simplicity, which I appreciate. I need terminal multiplexing, but I might use shpool here and there.