That capacitor is a snubber cap for RF spikes from coils. It is not smoothing the rectified waveform.

As mentioned earlier, you could have one leg of the bridge out. If you had an oscilloscope, you could look at the waveform and see what is going on.

Replace and report back. Also, double check the wiring to / from the bridge.

Great questions. I'll briefly elaborate and maybe others can chime in.
-Rectified AC has similar potential as regular AC with only positive waveform spikes. In theory, the positive peak potential is the same, but with less downtime between the waveform peaks. I'm not sure how this changes coil response. My guess, is it makes the coil respond faster.
-Without a scope, I would use the frequency counter on your digital VM to see what the waveform frequency is measuring. It should be double the line frequency (60HZ would be 120 and 50HZ would be 100).
-Switch gaps cause the RF spike. On SS games, it is not the low voltage switch contacts, but flipper switches and EOS switches. In EM games, all switches.
-RF Spikes cause noise in the power lines and without EMI filtering on the power line, it will radiate to other electronics. In SS games, it can also cause issues with low voltage electronics, microprocessor and so on. That is why there should be some RF filtering on MPU boards via LC and RC networks on the supply voltage lines.

Just order what PBR or Marco has. I use Mouser all the time, but you will drive yourself nuts trying to pick out the best cap if you are not familiar. Also, this should NOT be a polarized cap.

One more thing I thought of, most bridges have the sine wave symbol for the AC (input) and then the + / - for the rectified output. Do you have your wiring correct?