Troubleshooting UPS failure
It very well may, be that this particular unit is "advanced enough," to just not do anything once it senses that the batteries are beyond dead! There are many other problems, but dead batteries are almost always the problem!


  I don't really want to go out and buy new batteries for this unit if I
  haven't confirmed it works.


I have repaired a good deal of these types of units. The next most likely problem is that the "voltage sensing circuitry" is either bad, or is not designed to work with the power (voltage, frequency, etc.) that you are supplying it while you're trying to test it.


  What can I check before I start ripping the thing apart for connectors
  and inductors?


So these are the two main things to start with. However you should first research exactly what model it is, etc. (And post it back here so we can better help you!)