If you let the UPS stay plugged in for a few hours, then unplug it, does it stay on?
Then, try that with a small load, like a desk lamp, or something.
If the battery is bad (and was not detected as such), it should seem fine with no load, but drop quickly with a small load.
If the small load is OK, then, after it think it has charged up, try unplugging it with the PC on, just sitting at the desktop, or in the firmware interface, and see what happens.
If your PC's PSU has too low a hold-up time, there could be problems. Mine, by reviews, doesn't quite meet ATX requirements, FI. The UPS generally takes several milliseconds to swap to battery, anywhere from 4ms to 12ms seeming to be common. If it takes 10ms, but your PC can only hold out for 8ms, then there goes the PC, even with a fully functioning UPS.
Have you tested the UPS, before now?