Personally, I'd rather see the messages fixed. A long while back they went to the trouble of changing the game chat so that you see it "on top" even if you aren't in the game. So theoretically everyone should see the message regardless. But, if you add anything to the chat while not in the game, then you don't see that added stuff when you go into the game itself. It's as though the "on top" chat is a different one.
Just last night the chat of a (turn-based) game popped up while I was in the forums, and it wasn't a personal chat, I saw the game name at the top of it. And yet, nothing from our chat done in the game itself was showing. It was like it was a whole new personal chat instead. So now, looking at the chat in the game itself, only his side of it shows and none of my responses are there. (Which looks ridiculous.) I have to assume it's because he had the game open. Then my responses start showing up towards the very end, because that is when I opened and looked at the game, so now my chat was "within game." But almost all of my half of that conversation is missing, because I was outside the game at the time.
That seems quite ridiculous. And it also means parts of chats end up missing from the game. If the chats would work properly and everything for a given game carried through, both directions (i.e. inside the game carries to outside and vice versa), and worked properly, then other players not seeing your stuff wouldn't exist to begin with. I get that if they haven't taken the invitation yet they wouldn't be part of that chat yet, but if the chat truly "belonged to the game," then they should see it when they join. (Of course, making what was prior to even joining the game show to a new person might be very difficult. So your idea would come in handy as well, though it possibly could be just as difficult.)
/rant about chat
