AMI Model A Jukebox repair

Sounds like a couple things are going on. First of all, the gripper arm and clamping mechanism are notorious for getting gummed up with old, dried up lubrication. The fact that you can turn the knurled shaft on the back end of the gripper motor and make it move is a good sign, but I'm guessing your fingers were probably bleeding after doing it a couple times:). I would remove the gripper, disassemble, clean and relube. It's not a bad job; only 4 screws to remove. I would take a bunch of pictures first though, to make sure you realign the parts properly. The alignment of the gears can be a little confusing. I have an A that I recently restored if you need photos etc. After cleaning and lubricating, the gripper should move very freely.

Regarding the record not being returned to the rack, check the microswitch at the base of the tone arm. It's probably red in color. The tone arm itself should have a little paddle/lever at the base that pushes against the switch blade at the end of the record. There is also an adjustment screw there to fine tune when exactly the needle will lift and return the record to the rack. Since you said you can reject a record from the momentary button on the control box, the circuit has power. The switch I'm referring to also plugs directly into the smaller control box under the turntable. Make sure the plug is properly seated.

One other question - Does it accept coins , or is it somehow on free play? You mentioned earlier that it makes a noise when you press a selection button. Do the little levers on the pin bank under the record rack flip up when you make a selection?
Carl.