Perhaps a third option that accounts for the preference of some players to win by "battle to the death".
Before start of game, players choose between:
[editted to add: this would be selected "per-player", not as a game-level option that all 4 players need to agree to]
1) [Default] auto "claim win" is enabled; "claim win" is performed automatically by server as soon as a player would win even after resigning.
2) Auto "claim win" is disabled. This means players will not get "claim win" option. In fact, in the spirit of "non-random outcome" and eliminating ethical choices, if this option is chosen, players MUST checkmate/stalemate final opponent for the win (or opponent must resign). Regardless of point lead, if a player is using this option and either flags or resigns while in the lead, their final score is set to the smallest of [points at time of resignation] and [winning player's final score minus 1].
Naturally, each player's selection should be evident upon the start of the game; so in situation where one player establishes dominating lead, all players can optimize their strategy according that player's selection.
After viewing one of "dubiousskills" excellent videos, I felt it served as a perfect example of "auto-claim-win" improving the game.
The video is titled "4 PLAYER CHESS COMMENTARY (100+ POINTS) 1745!!"
Link: https://www.youtube.com/watch?v=apuJkggoFy4
Dubiousskills' channel (highly recommended!) : https://www.youtube.com/channel/UCntDXISnx7OSZZ4fcueapzg/videos
I recommend watching the whole game. But my emphasis here is on play from the time green captures yellow's king (at 23:30).
After this capture, there are three players left: green, red, and blue. Green quickly parlays his king-capture into utter dominance; leading both blue and red by 10+ points, 2 queens, and positional soundness (e.g. king safety, pawn structure, etc). By the 25:00 mark, it's clear that even if blue and red 100% conspired, taking green down would be a long shot. However, while first place is almost certainly out of reach, blue and red have a dead-heat battle for 2nd: blue has a +11 point advantage, while red is up in material and has a better position.
This is the point of game where I believe the auto-claim-win feature would begin to objectively improve the game better for all three players. Green's strategy becomes very clear:
1) maintain (or extend) point-lead
2) weaken blue & red until one of them can be checkmated.
Meanwhile, by this same 25:00 mark, blue & red are clearly in a "battle for distant 2nd". Both should realize "Green will be aiming to checkmate one of us to secure the win." With "auto-claim-win", they would also realize "Whoever green doesn't checkmate will get +20 points".
As such, blue would have 2 realistic ways to beat red:
Aggressive: extend the 11-point lead to a 20+ point lead over red, such that even if green checkmates blue first, red's +20 point bonus won't be enough to catch up. Look for ways to exchange rooks/bishop with green to boost lead.
Defensive: With an 11-point lead, red's only hope to win is if green decides that it's easier to checkmate blue than red. Defend blue king at all costs; and, if possible, try to make an attack on red's king more attractive to green by weakening red's king defenses. Perhaps even sacrifice any material that isn't defending blue king in a way that opens up red's king. Make the "mate-red-for-win" as lucrative to green as possible.
Red meanwhile has two similar strategies:
1) Long-shot: super aggressive aim to gain 31+ points over blue before green can mate red (maybe possible if a pawn or two can be promoted and green allows a lot of piece exchanges).
2) More realistically: use material/positional advantages to ensure blue is mated first (there's really not much hope of gaining the 31+ points on blue). Defend red king and make "mate-blue-first" more attractive to green than "mate-red-first".
Skip to 32:00 to see how the lack of auto-claim-win sours the game overall. Red winds up doing a worse job than blue (IMO). Red does overcome blue's +11 point lead, but only to pull ahead by a meager +4 points (about half of the +31 needed to justify an "aggressive" strategy). Worse, in working to gain these points, red has also failed on the second strategy: neglecting king safety, leaving red open to the quick&easy mate from green at 32:30.
And this is where things go bad (IMO). Green has to make an ethical decision: to claim win (giving blue +20 and second place), or to play to checkmate blue (giving red the win with +4 points). I don't think it's good sport to force green into making this decision; and I *really* don't think it's good sport to leave red&blue's fate in green's hands. Frankly, I could see either red or blue feeling a little bitter/cheated, regardless of whether green chose to claim-win.
However, the lack-of-auto-win played a role in red's decision. It introduced a random variable: since red couldn't count on green to auto-claim win, allowing blue to retain even a sliver of a lead would lead to risk that green mates blue, doesn't claim-win, then mates red.
Anyway it plays out, I think the lack of auto-claim-win makes the ethical choice of "claim win, yes or no" a dominating factor of the 3-players-remaining dynamic. This is a problem, because that ethical choice has nothing to do with how skillfully either of the other two players choose to play.