Yep! Use copy-pipe-and-cancel instead of copy-pipe-no-clear.
I don't think tmux has any, like, OS-aware anything. It can copy to the system clipboard, but only by asking your terminal emulator to copy. To work the way you expect, tmux would need to know to use xclip on Linux and pbcopy/pbpaste on macOS or whatever. Which would be pretty nice! That would be a great, easy feature that would cover ~99% of users. But it seems like it deliberately doesn't want to (?). I'm speculating there.
I don't think tmux has any, like, OS-aware anything. It can copy to the system clipboard, but only by asking your terminal emulator to copy. To work the way you expect, tmux would need to know to use xclip on Linux and pbcopy/pbpaste on macOS or whatever. Which would be pretty nice! That would be a great, easy feature that would cover ~99% of users. But it seems like it deliberately doesn't want to (?). I'm speculating there.