Version 4.0 of AI War was the first version that even supported OSX, so that's definitely not what you're running into. The version number you see there is the version number of the Unity 3D engine, not of AI War itself.
I'm still running OSX 10.6.6, so I'm installing 10.6.7 now to see if perhaps Apple broke something there. But that would be an issue for the entire Unity 3D engine presumably, and I think I would have heard something about that, so I'm not expecting anything -- will let you know once that finishes, though.
It really does sound like it's not properly installed, to be honest. I've had... some interesting experiences with the Steam client on OSX. Often it tells me that it's verifying the cache files, but it obviously does no such thing because I've deleted all the files and it doesn't notice. Things like that. So I'm not overly trusting that it's actually downloaded everything properly.
One thing you could do to circumvent this is to actually install AI War 5.0 via our direct installer, and just point it to your /Users/JohnDoe/Library/Application Support/Steam/SteamApps/common/ai war fleet command/ folder as the installation target. That would overwrite whatever the Steam client downloaded with a version that's definitely good, and then you're set to run it from there with the steam client, etc. Assuming that Steam downloaded all the expansion images and music and such properly, you wouldn't need to download and use those installers separately.
The other thing you might try is running the file /Users/JohnDoe/Library/Application Support/Steam/SteamApps/common/ai war fleet command/AIWar.app directly and see what happens. It could be that it really has been downloaded properly, and the Steam client is for some reason having trouble launching that. I only saw that once, roughly a year ago, so that's less likely but it is easy to do.
I'll post again once I get up to 10.6.7, and let you know if the game runs under that. It should, but it hasn't been tested since I hadn't noticed it was out.