The only question is whether the time between power starting to drop and the power going completely is enough to flush pending writes (possibly reordered), while also not having any RAM contents get corrupted
That's easy: No. When power starts to drop, everything is gone at that moment. If the disk is writing at that moment, the unfinished sector gets corrupted, and maybe others. UPS for the computer and disk together helps only a little against corruption unless power drops are almost always shorter than the battery time, or it automatically shuts down the computer before getting used up. You may be better off if the computer loses power immediately, and only the disk gets the UPS treatment.
it really shouldn't be necessary to use a UPS just to avoid filesystem/LVM corruption.
Perhaps, but it is. (What is this "should"?) The file system doesn't matter near so much as you would like. They can be indefinitely bad, but can be no more than fairly good. The good news is that the UPS only needs to support the disk, and it only needs to keep power up for a few seconds; then many file systems are excellent, although the bad ones remain bad.
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