Essentially the reverse light circuit goes like this ...
Battery positive -> main fuse -> ignition switch -> ignition switched circuit main fuse -> turn/back fuse -> reverse light switch -> reverse lights -> ground.
You cannot verify power & ground continuity at the reverse switch connector - you can verify power and switch conductivity - the "non 12v wire goes to the lamps and through them to ground.
If you have power at the switch and conductivity through the switch when reverse is selected, then you have a problem "down stream" from the switch - either a broken wire, a bad connection or defective lamps.
The trailer wiring "tees" into the factory wiring and will not normally affect the reverse lamps (at least not unless there's a short that blows the fuse.)
93 - 1.3 Suzuki Swift GLX
98 - 1.8 Mitsubishi Pajero iO
98 - 2.0 Suzuki Grand Vitara