I have a feeling that transformer is also there for isolation so the strobe doesn't introduce noise back into your mains supply.

The trigger voltage on the xenon strobe lamp can be a few kilovolts and ignition draws high current. It will generate unwanted EMI which you don't interfering with the game and other household devices.
The rectifier board and big cap on the SDB are not there to filter voltage spikes.

Inspect that transformer, it's likely the winding has broken from one of the lugs - remove some of the paper covering if you need to get a better look.

Consider why Bally would add the expense of an extra 1 to 1 transformer on machines destined for the local market only. There must be a reason.