The cap/resistor form a timing circuit called an RC network (guess why!) - it's there to provide a minimum pulse time so the coil reaction isn't too short.

Pull the connector off the input and try grounding the input pins on your mpu board that control each pop see if it's better - if it is, you have a bad ground going to your activating switches.

The inputs for the special solenoids are connector J18. Don't dwell on the pins too long or you'll blow a fuse/transistor - just a touch like you were testing the coil under the playfield via the transistor tab.

There could be cracked headers on your mpu as well - if they've never been resoldered, might be time. Suck most of the solder off first, don't just glob new/more solder on.

You sure you have the correct value cap in the circuit and it's pointing the correct direction? Sometimes the old caps are marked with the line meaning +, not -. Even new ones are sometimes different.

The lack of a tie point for the cap/resistor connection has no bearing on the circuit, it's just a convenient tie point.