Power off the machine and then power it on again. Manually push in the reset relay and see if it will start the reset process. If not, look at the switches on the reset relay to see if they're clean and adjusted properly. Also, check the coin chute switches to see if they're stuck in the closed position...

Looking at the Little Joe manual about "General Game Operation," it looks like the machine isn't advancing to the next step in the reset process. Does the score motor start turning when you press the start/credit button? Does the total play meter advance when the reset process starts?

After the reset relay is energized, the score motor will start running and close a series of score motor switches, which will keep the score motor running to continue the reset process.

There might be a dirty or mis-adjusted switch on the score motor. Take a look at score motor switch 1E with the white wire, red/yellow wire and the black/white wire. Make sure this switch is clean and adjusted properly...

https://www.ipdb.org/files/1460/Bally_1972_Little_Joe_Manual.pdf

To confirm, when you manually push the reset relay, the machine goes through the complete reset process, all the score reels reset to zero, the ball kicks to the shooter lane, and it's ready to play the 1st ball?

When there are credits showing on the credit unit wheel, and you push the start/credit button, this will send power to the credit relay coil, which will close a switch that will send power to the coin relay coil. Then a coin relay switch will close and send power to the reset relay coil. So, this might be an issue with power getting to the credit relay unit coil. Try to manually push the credit relay and see if this will completely reset the machine.

Yes, the dual score motor switch 1E with the red/yellow wire should be open at the start of the game. The other side of the switch with the black/white wire should be closed at the start of the game.