Clean Jones plugs to start. Clean the L, M, N relay switches and check gap on the ones that activate the chimes. Both of those are quick and easy; if no go someone with a schematic can help more.

The chime coils all share one common wire, and each also has it's own wire which comes from the corresponding relay. These wires are attached with spade connectors, and are often reinstalled out of the proper sequence. It's just a matter of connecting the proper wire to each coil to get the correct tone for each point value. The highest tone is tens, the middle tone is hundreds, and the lowest tone is thousands.

That’s exactly what happened on my Jumping Jack when I got it. The spade connectors on the 10s chime and 100s chime were reversed and it resulted in a similar way as the OP’s.

Cleaning the chime coil sleeves and plungers usually solves this problem.

Not doubting you but you mentioned a couple of things that were going on that sounded very familiar.

When I manually activated the 10 point and 100 point relays, they sounded the chimes but what it was really doing was sounding the 10 point chime only.

When I manually activated a rollover for 100 (or 500 points in my case) it weakly chimed but I had to raise the playfield to notice that once again it was the 10 point chime only that was trying to sound off.

You also mentioned that the 1000 point chime hardly ever goes off and mine never did due to the fact that there was no path to it because of the backwards connectors.

Might be worth a shot to switch the spade connectors on the 10 and 100 point chime coils to at least eliminate that possibility. When I did it out of desperation it was a huge relief to something that was really aggravating. Hope that’s it.

if you don't find anything with the chime wiring, the only real difference between you manually operating a relay and the game circuits doing it is you are almost certainly holding the switch on the relay closed longer and you may be making a different/better connection by tilting the armature plate.

none of the below would explain the wrong chime powering (assuming it's not just when the digit is rolling over from 9->0), so if that's definitely happening, ignore the following.

tmi alert
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there's hold-in switches on the relays that keep the relays powered until score reel switches open the circuit. E.g. on the N relay, there's a switch with WH-BL wire and the SL-WH-RED wire going to the coil that is the hold-in switch.

look for the WH-BL wire on the 10's reel on player 1 ... usually it's an EOS type switch. You should have 25V between the WH-BL wire and BLK wire on the coil when the reel is not doing anything.

alternatively, measure resistance between the WH-BL wire and RED-WH wire on the same switch on player 3 tens reel. You should have almost zero ohms.

also on the N relay is a switch with MAR and RED+WH wires. That's the one powering the chime. Make sure it closes quickly/reliably.

it's basically a race condition. The reel coil and the chime are more-or-less firing at the same time via different switches on the controlling relay. If the coils are wrong or switch behavior doesn't make the chime plunger do it's thing first, the reel will unpower the circuit before the chime has ... chimed.

Awesome. Glad you got it fixed and working right.