I assume you’re trying to do a workout in ERG mode, which should work. To be clear, ERG mode is when the the trainer resistance is controlled by Zwift to match a certain wattage no matter your cadence. This should not require any gear shifting.
On the other had, if you’re free riding around in Zwift, do you feel resistance changes when you hit a hill? This will require gear changes to compensate for power just like in real life when you’re outside.
If you have a PC w/ANT+ dongle, give it a try. If that works, then you know it’s not the trainer! Well, unless something is whack w/the trainer bluetooth.