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I don't think they're confused by times like 1pm.
At least for my brain, 12pm and 12am are the sticking points.
As you note, pm is Latin for after noon, yet we call noon 12pm. Noon isn't anymore after itself than it is before itself. Neither makes any sense.
With 12am, we generally seem to think about midnight as the end of the day, even though it's really the start of the new day. The Latin isn't confusing here, but the numbers get real weird. We start the day counting at 12:00, go up to 12:59, and then reset the count to 1 an hour in? Our 12h clocks are split between being 0-indexed, and a weird variant of modulus 12.
I'm clearly overthinking things, but I don't always immediately remember which 12 is which. Latin doesn't help.
With 00 it's clear which time we're talking about, and which calendar date it's part of. It's also the easiest way to sort out which 12 gets mislabeled what.