Radio assist is something that gives you magic warnings like "missile is inbound." It's nothing to do with using the radio.
There are four radio transmit keys in the A-10, forward, mid, aft, and hot. In real life they correspond to pressing the MIC hat on the throttle forward, down, rear and speaking into the microphone without pressing anything respectively.
All communication with ATC is done using the VHF AM radio which uses the forward push of the MIC hat. The F1-F12 type menu that pops up should indicate the radio being used. Considering you are able to hear ATC it appears the VHF AM radio is set correctly for reception. For transmission, make sure you are pressing the control bound to HOTAS>HOTAS MIC Switch Forward to initiate the F1-F12 style menu.
The ATC is programmed to act as if you requested a phase in the control path if you physically do that phase. It minimizes the chance of being stuck in a phase. It's not normal operation.