Not sure about the lights not working, we'll get back to them......for now there is another issue you will need to address....you put in new batteries because as you noticed one of them went down hill...from the looks of things they didn't "Convert" to 12 volts....what they did was put in a 12 volt "tap".....as you found out, that will kill one of the 2 batteries, with only "tapping" off one of the batteries you make that one work harder than the other, so they get an uneven charge put back on them, causing problems. You can put 24 volt bulbs back into the light sockets and hook the light switch back up to 24 volts and have no problems....or put in a 24-12v converter and have a true 12 volt supply that won't harm your 24 volt system.
Now back to the lights....you can pretty easily trouble shoot your problem with the way they used the light switch they did....it will have a single power "in" point and should have 2 power out points...one for head lights and one for running.......if you feel frogy, you can un-hook the wires, use a test light and find the power wire....put it back, now on the other 2 wires run a temp positive wire to one of them at a time, if the head lights come on than go to the next wire, hook it to a battery and see if the rest of the lights come on, if so than the switch is bad if not than you have a ground problem with the lights.....more than likely.