How to mute monitors automatically
You should be able to find monitor muting as a default or configurable setup on most radio consoles. Make sure you looking at sound boards that are specifically marketed as radio consoles. Something like this Wheatstone Audioarts analog console or this Lawo digital console. If not specifically called out in the features, you can also look for consoles that can trigger on air lights, as those contact closures can also be used to mute the monitors with some simple, inexpensive outboard gear. A Mackie or other inexpensive all purpose audio board isn't likely to have either feature.