I have a bravo III and recently had a coupler issue and agree with most of the above- with the drive still on and turning the prop you should be able to open the hatch and look down at the input shaft and see if it is turning when the prop is turned and in gear.-you might have to put some color on it to see if it is moving. If not, then the effort to remove the drive is minimal- 6 bolts and a couple of strong backs or in my case, a dolly I built and off it comes. Then you can see what is going on with the shift cable as well. Then you can look up into the coupler as described and see what is going on. The Bravo III are pretty good drives I'm told, one of the best. I was just putting when my coupler went and had just launched. I had struck something submerged a couple of months before and assume that was the set up for the failed coupler. The bravo III do not have prop hubs to spin- I don't know if other drives have similar couplers or not. One thing I do know, is if you hit something you sure don't want the drive to take the hit. I think I spent 1200 bucks to have someone do the coupler as I had no way to lift the motor. I currently have my motor out and rebuilt my transom assy from a leak and getting back together shortly. Will have someone install the motor ( I'm 14' off the ground to clear the sides of the boat to get the motor in) and have them align the motor as I don't trust that I can do it 100%.
Good luck, please post the findings!