Steem actually does this a little with its memo keys, but transferring even a tiny bit of money isn't the best way to send private messages. The three second transfer time could also be annoying for messaging; though a queue, like SteemPeak has, could solve that problem.
Definitely a lot of possibilities in this area.
You're right, I meant operations rather than transactions. I don't think three seconds is actually that long of a time especially for private messaging for two folks. If you're talking about something like a group chat that spits out multiple comments a second, you're definitely right on that. I guess second layer solutions (like a queue) can solve those issues.