I just got an email that a new version of Gladiabots is out and that I can download it. But I used the itch.io app to download and install the previous version. So...
Does the OSX itch.io app automatically download and install updated games?
Yup, if you have something installed and the app is running, it will automatically download and install updates.
What leaf said — when you start up the itch.io app again, you should see a download for Gladiabots being queued — when it's done, you'll be updated to the latest version.