I agree that there are problems with having a single page tablet. I ran into it when I was trying to take a snapshot while playing an emote. I thought that was what the pin button was for, but it does not seem to do anything.
A possible solution would be to have tear able tablet pages. if you wanted to have chat always open, you could just tear it off and it would become a separate window that would stay open while you then could do something else on the tablet. Or tear off the emote window to take snaps while doing emotes.
I know this sort of clutters up the view in an HMD but once there are many tablet apps available, I think this issue will happen a lot.