Hm. I read back over the thread, and I see I didn't read this post carefully enough:
13CARDS;417102 wroteIf you removed Player 3 from the scenario and made it a heads-up pot between Player 1 and Player 2, and Player 2 ended up with the winning hand, Player 2 would eliminate Player 1. Definitively. There is no scenario, using the chipstacks, blinds and antes I described in the OP, that Player 1 could ever eliminate Player 2. If Player 1 ended up with the winning hand, Player 1 would "double up" and Player 2 would be left with 1,000 in chips.
I... agree. Heads up, P1 posts the BB (12,000), pays the BBA (12,000) into the pot and now has 48,000. P1's shove can only net twice their bet amount (stack 48,000 + BB 12,000 = 60,000) if P2 calls. So yes, a player with a shorter starting stack could eliminate a player with a larger starting stack if they are heads-up. This is surprising and counter-intuitive.
However, the BBA has to come from somewhere, that somewhere is P1's stack. If P1 and P2 are knocked out during the same hand by P3, P2 is still out before P1. It doesn't matter that P2 potentially could have knocked out P1 if they'd been heads-up, that's not relevant when it's not heads-up.
The nuance is that antes are not counted as part of one's bet, they are dead money. Blinds are counted as part of one's bet. To that end, I was wrong when I said "There's no fundamental difference (beyond who pays how much) between SB+BB, SB+BB+Antes-from-everyone, SB+BB+BBA, any other arrangement." There is a fundamental difference between blinds and antes; blinds count towards your bet, antes do not.
In a balanced ante situation where everyone antes the same amount, the effect of not being able to count your ante as part of your bet is negated because your ante is already matched due to the balanced situation.