The gear marked #3 in the diagram should turn freely in one direction, but not in the other direction. If it won't turn in either direction, you've found the problem. There's a one way clutch or bearing that will allow the gear #3 to spin freely in one direction, but the opposite direction is what starts the engine. The starter gear stays meshed with gear #3.
Does the starter spin at all? It may be spinning in the wrong direction. Ask me how I know 😡.
If you can get something into the hole that houses the gear #3 besides your finger, such as a screwdriver, try to spin that gear using that. If you still can't get the gear to spin in either direction, you'll most likely have to replace the one way bearing.