Sounds like your hold relay (R) is not engaged. You have the manual? If so, follow the start-up sequence...
*Coin or replay button actuates the start (S) relay
*When the S relay is pulled-in it does multiple things such as starts the score motor, reset score reels, reset ball count, and subtract from the replay unit. (The V relay will also get engaged).

Can you get to that point on startup?

Not sure on the wire from the photo. Can you trace it back to where it belongs and can double check that way? I think you are probably correct in your assumption.

Can you manually adjust the ball count unit (backbox) with the game off and reset it manually? Should advance 5 times and then reset back to zero position. Try a couple of times and see it moves smoothly or if it's gummed up.

That's good. The one of the left is the replay unit and should only add or subtract one at a time. Basically, how many credits are on the machine.

What do you mean here? If it's on ball 5, it shows ball 4 after you hit the credit button? Can you snap a photo of the ball count unit in the zero position?

Ball unit looks correct in the zero position. Can you make sure the switches I circled switch states as you advance the ball unit? These should open or close at some point. Some will transition the first time you advance it. The top one should open when you can’t advance any more.

The teeth are correct. This is made so you can’t advance too many times. The last advance should be on the last rivet.

If all check, then the ball count unit is good.
534BEAA7-91F0-40A9-A71E-94D205163951 (resized).jpeg

When you start a game does the R relay just pulse and not stay engaged? Does the scores reset and motor stop spinning?

Here's the path for the R relay to engage. Check the switch that I circled. It's a make-break switch on the S relay in the bottom of the game. There will be three blades with two contacts. It's the only one on the S relay. The wires colors are (White with a Brown tracer) and the other one is slate. Make sure the switches are transitioning when you manually move the relay.

Rrelay (resized).JPG

Is the game now working besides the buzz?

Those switches are on the relays not the score motor. Those switches circled are the only ones that cause the score motor to turn.

Sounds like one of your drop targets is showing a closed switch. Put them all in the up position and see if the problem goes away.

If not, then you need to check the switches on the drop target.

If they problem goes away, you can hit one target at a time and see which one is making the motor one continuously and adjust that switch.

If it is the total play meter, it can connect in a few spots. I highlighted everything that connects to the Brown & White wire. Also, sometimes people don't hook up the total play meter.

loose (resized).JPG

You're coils look good to me. What changes did you make before the fuse blew? I would put in a circuit breaker and see if you can tell what coil is being pulled in. It might be more difficult than that, but that's a good starting point. Be careful with the 4 drop target reset coils. They are 120V.

Shameless plug, but I do sell circuit breakers on my site .
https://troxelrepair.com/product/circuit-breakers/