If you are actually able to engage the gears, release the clutch, and the car does not move, there is most likely something wrong with the clutch. It could be several things. A bad pressure plate will not allow the clutch disk to firmly press into the flywheel. Incorrect installation or an old clutch assembly could be the cause. Also a worn out clutch disk will cause similar issues. If it is worn down, the friction material may no longer contact the pressure plate and flywheel enough to let the engine and transmission mate.
I can’t really see any of the shafts inside being damaged to the point where the transmission could have gears engaged and not allow the car to move without making quite a lot of noise. It would be very noticeable. That is why I would believe it to be the clutch.
Have you tried starting the car in gear? If the car initially moves while you’re starting the engine while depressing the clutch with the car in gear, the slave cylinder would be the problem. If the car doesn’t go anywhere even if the gear is engaged, I’d say it’s a bad clutch.
Also, how have you determined the clutch is good? Just because the master cylinder has fluid in it and the clutch pedal can be pushed to start the car, doesn’t mean the clutch is good. It does mean the switch under the clutch is working correctly however.