Dubious. Even weak batteries probably have enough juice for an actively correcting UPS to protect things on mains.
APC SmartUPS, so they should have been OK. Quite an old unit; the current (replacement) batteries are about 4 years old. Almost time for a new set? I have a newer (but smaller) SmartUPS, which I'm using right now.
I find that heat (bad ventilation), and fan bearings, are the only things that killed the PSUs I've seen.
We have a real problem with dust. I don't know if it's unusual, but I do get quite large
dust bunnies (not quite the same size as in the link, but occasionally close) around the back of the computers.