I was just wondering why this site doesn't have a chess program in the chat rooms, so users from around the network could come in and play?
I can think of three reasons why:
Stack Exchange's primary purpose is to be a network of Q&A sites. All features, including chat, support this purpose. I don't see how adding a playing arena would be beneficial to this, except maybe to enlarge our audience.
The Chess SE chat isn't really active at the moment. Even if this would be implemented, there's no guarantee that people will actually use it instead of the thousands of other chess playing sites on the Internet.
The Stack Exchange developer team is very busy. Until recently, they didn't even have time to properly maintain the Android app, which is quite a thing in the mobile-first world we're living in now. Chess SE is a small community, and we should count ourselves lucky that we have one feature specially developed for us (the replayer).