Agree with measuring the voltage first. Heck, the voltage is going to
be low. Why not just determine what the battery voltage is supposed to
be and if it's 12 volts, put each battery on a car battery charger for
a few days. Check the voltage, did it go to 12 or more? If so, put
them in and see if the ups will come up.

I'm only familiar with commercial large scale UPS's, know nothing
about these little guys. What happens with these little UPS systems
when the battery voltage gets low after a power outage? Do they just
shut off, switch to raw power or what? Your system may just be
experiencing what it thinks is a low battery condition and it does
make sense that the circuitry would be such that if there was no
battery voltage, the unit isn't going to do anything but shutdown. It
also may require some battery voltage to even turn on.

Consider the car battery charger idea just to see if the batteries
will take a charge then try them.

The stored energy in a car battery is much greater then the 60ah
mentioned, go by cold cranking amps and your are looking at 500 plus
amps of power instantaneous. If you are the adventurous type, put the
daggone ups in the driveway, run a couple number 10 gauge wires to the
ups, run a 100 foot power cord to the ups leaving it unplugged from
the house or garage, plug it into the UPS, and then go back and plug
it into the garage or house. Get your binoculars and try to see if any
lights are on on the UPS. Also maybe consider waiting until night time
to do this just to get the benefit of any short circuit there is, or
July 4th. Sorry, just having a little fun but not serious, best to
play it safe.