So, power comes on. Game kicks a ball out. Game kicks a second ball out. Then there's some kind of sound callout. Is that part of the game's attract mode? I also didn't see the kicker try to kick the third ball out, nor did I see it lock on. Can you get some video of that?
The initial kick when you turn the power on reminds me of a Taxi I had where the bell coil would fire right when the machine was powered on, before the CPU had even booted. Turning the machine off and then right back on again would fix the problem until the machine had been off for a while.
https://pinside.com/pinball/forum/topic/taxi-bell-rings-on-power-on
It also makes me sad that I never wrapped up the thread with my solution. I hope I can remember it. Also, anyone can feel free to correct me if I'm wrong about something. (I'm probably wrong about something.)
The general story is that there's a board-wide signal that limits when the coils can fire. This signal acts in conjunction with the signals sent from the PIAs (which are, in turn, driven by the CPU). There's a timer chip that's designed to wait until the CPU is in a stable state before opening the gate for these coils to fire. I think this is the "blanking signal" that you see people talk about. Or maybe it's related to it.
Somewhere in the PIA-to-Coil equation is a logic chip that examines the blanking signal and the signal from the PIA. If both of those are in a certain state, then this chip sends a signal to the transistor that drives the coil. If that chip is faulty, then neither the blanking signal nor the PIA's signal will matter. The coil will just fire as soon as that logic chip gets power.
I have to go right now, but I'll be back later. For now, scrounge up the manual and grab your logic probe. Find a nearby coil (schematic-wise) and check the pin from the logic chip that drive's that coil's transistor to see what it does when the machine is first turned on. I'm willing to bet there's no signal at any point during that process. I'm also willing to bet that the pin leading to the transistor that drives the kicker coil is high for a bit of time as the machine first turns on.
Please note that all of what I just said could actually be completely wrong. I'm still kind of new at this stuff. Hopefully it helps!