Hmm. How have you determined the belts are turning when you were turning the ignition key? If you're hearing the engine turning over on the starter motor with the normal Rrr-rrr-rrr sound, then yes the starter motor is indeed working. If you were hearing a "click", or a "clickity-click-click-click" sound, the engine is not turning over and you've got a problem with the starter circuit.

Since you later seem to say the engine is indeed turning over on the starter motor, I'll go with that presumption. Which makes the diagnosis much more difficult.

The trouble with hunting one of these down is as soon as it starts, you can't locate it. You can only hunt these sort of things down when the engine will not run.

There's two areas to search, ignition and fuel supply. You can check the ignition readily enough when it won't start. Just pull a plug line and check for spark. If you've got spark and the engine still refuses to run, you can be pretty sure it's in the fuel supply side.

Almost certainly it's going to be an electrical failure. You need the wiring diagram to chase this down with. Somewhere in a controlling circuit electricity isn't making its way through the path. It can be a dirty contact (common), lose connection, corrosion, chaifed wire, etc. But you need to know where you're looking to find it.