I think the point on the connector isn't concerning the flipper necessarily, but if those connectors are cracked.... the other ones probably are too. There's really no point to rebuilding the connector pins in the housings if you're not also going to (minimum) resolder the cracked connections on the connectors (really, should be replaced, too). Quench probably means the solenoid driver board's j2 connector not the rectifier board as you were talking about earlier.

The power delivery from the rectifier board to the saucer is through a chain, if you have other solenoids working on that chain, the power it getting to the coils. Now, if there is only one wire on each of the saucer's coil's lugs, it might be the last coil on the chain.

I take it you don't have a multimeter because you can tell in a second if you have power at the coils or not with one. My suggestion is to go get one, for what you need to test a cheapie from walmart or harbor freight will do. Black lead on ground braid in head or cabinet, red lead to either of the coil's lugs, setting on DC volts 200 range (on most meters). You should see between 44-70 volts (don't worry if it's really high, that doesn't really matter).

Tug on the wires as well gently (power off) - there might be a break in them that's really subtle and only occurs sometimes (which seems to be your core symptom.... sometimes it works and sometimes it doesn't.)

You might still have issues with the solenoid board connector it's common for them to need attention, on both the mpu side (lower left connector) and the solenoid driver board side (lower right connector). Just because some of the signals work doesn't mean they all do.