@jumpyknight8 your suggestion was supposed to prevent the winner from determining the rank of the other players. But it doesn't. It only makes it a bit harder for the winner to do so, at the cost of making the whole game considerably more complicated. It also incentivizes anticlimactic strategic resignations at the 3-player phase of the game, and creates possibly unwanted side-effects like punishing harder the passive play. Based on that I can't give a good rating to your suggestion!
Suggestion: Claim Win points based on remaining material

@jumpyknight8 your suggestion was supposed to prevent the winner from determining the rank of the other players. But it doesn't. It only makes it a bit harder for the winner to do so, at the cost of making the whole game considerably more complicated. It also incentivizes anticlimactic strategic resignations at the 3-player phase of the game, and creates possibly unwanted side-effects like punishing harder the passive play. Based on that I can't give a good rating to your suggestion!
Yes

The objective of this game is points. Likewise the objective of standard chess is checkmate. Consider this analogy: In standard chess a game ends immediately after a king becomes checkmated. Nobody is interested about what would happen if the game was continued with one king on the board. Maybe the other king could be checkmated later on, possibly at the very next move. But nobody cares. You lost your king -> you lost the game, end of story. In 4-player chess the winning condition is: have more points than anybody else. The moment you achieve this, you can end the game immediately, and you are the winner. Nobody should care about what would happen if you had allowed the game to continue. The analogy is not perfect, because the win claim is not obligatory like in standard chess, but it's still a pretty good analogy IMHO.
It shouldn't be about points. It cheapens the game imo. But oh well.
I think that your suggestion punishes passivity a bit too much.
The best last move of Player-D could be easily calculated in advance by Player-B, so that he can be pretty sure about the outcome of his resignation. Actually, keeping with the spirit of the suggestion, the last move of Player-D could be safely automated, by auto-selecting the move that grabs the most valuable piece of Player-A available (or checkmating him if possible), to shorten the game by one more ply.
Also the precondition of capturing the leftover kings before the activation of the auto-claim mechanism, could allow the last remaining players to collude with each other with the intention of both outscoring the eliminated players. As an example Player-A could delay the capturing of Player-B's dead king, and first give his queen as a gift to Player-D, to ensure that his opponent will finish 2nd instead of 4th.
Yeah, I've been having second thoughts about Player D having a free move after the king capture, but it's probably necessary because of checkmate possibilities.
And the collusion is definitely a problem too. The best I can offer at the moment is a weak argument that you can already do this under the current system.