Update 1
Received a response about 14 hours later from Belkin Support.
The response listed several steps to test the unit, presumably to ensure that we eliminate any possible user-servicable issues before reaching the repair/replacement stage.
The process to try to reset the unit takes a minimum of 8 hours, since the unit has to try to recharge the backup battery.
Unfortunately, the steps did not fix the problem, and I've responded back to Belkin Support to let them know that the unit is still not functioning correctly and to ask what the next step should be.
I still think the unit will need to be repaired or replaced.
Will continue to update here as the situation changes.
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This issue does not involve any connected equipment having been damaged, so I cannot shed any light on that part of the warranty coverage.
However, I've used Belkin surge protectors / lightning arrestors for years and have never had any connected equipment damaged by a power surge. This includes a lightning strike that happened on exterior power lines immediately outside my house back when I was living in the States. An old computer not connected through the Belkin unit (instead connected to a "surge protector" but not a "lightning arrestor" which is apparently something different) ended up with a fried internal modem, but the system connected to the protector was fine. Something similar happened to me during an electrical storm on the Gold Coast a couple years ago, and the unit just make a "click" sound when the lightning hit (probably switching over to battery power), and everything was fine.
Bottom line is that I'm very confident about their performance, as I had terrible luck with surges and lightning strikes before buying their equipment and haven't had a problem since.
However, I've never had to put in a claim with them before now, so I have no idea how this process is going to go. We'll see!