Liquid, I do understand that my positions are necessarily personal and many will not agree. Wanted to share my views nevertheless, maybe it could help the collective reflection.
I said the 2nd player is the main loser because quite often, at least in the 2400+ games let's say, the 2nd player is in fact the one who messed up the 2nd stage (from the 3 stages), which is notoriously the most complicated and subtle. I mean, it's someone who could've (/should've) won, but couldn't because he made mistakes. In this sense he is the 1st loser. (You are usually 3rd and 4th because you played badly both the 1st and the 2nd stages...). At least that's my feeling. (cf. also https://www.chess.com/clubs/forum/view/the-real-loser-in-ffa)
I do not agree that "4th should be punished a lot" because far too often it's not your fault, especially when you're G and B is passive. And the strategies you mention are not always effective, in many cases you are simply not given this opportunity/time, if the sides are very good and cooperate extremely well.
"I think that there is nothing wrong with 0 for 2nd because it is only players who don't play to win who continue to team with their opposites, meaning that one could put such players in 3rd.
Are your concerns primarily aimed at those who foolishly seek to team to the end and have opposites who do not punish this?" : no, it's mainly to address 2 problems:
1) players who understand that they don't have anymore a lot of chances to win, so prefer to throw a game to someone helping him to win in order to be sure to be 2nd (and, in your configuration, not to lose any rating). It creates a wrong incentive I believe.
2) players who would like to leave a game and thus artificially shorten it, accepting to be 2nd, playing a kamikaze in order to accumulate points (trading, etc.), as it modifies the natural order of the participants at the end, and is quite unfair for the victim of such behaviours.
Basically if the 2nd = 0, it creates too many perverse effects. For instance if I'm SURE to be at least 2nd, it becomes positive expectancy to take bold risks (maybe you're 1st and win, or in the WCS you don't lose, being 2nd) or to trade.
This game that I just played is a great example of how 2nd = 0 works very well. https://www.chess.com/variants/4-player-chess/game/27692772
Red starts pushing forward into my position. At some point he must stop and let me consolidate so we can weaken green. But he doesn't he keeps coming so having the option to bail out works very nicely. It's too risky here. If I allow red to promote again he may continue the attack further and I can end up 3rd losing maybe - 15 and the fact that I'm stopping him promoting is making him more frustrated. There is no way back here and it benefits me to settle for 2nd. So at low rating levels this works very well as you cannot be sure what players will do. The question is whether 2nd = 0 would work for high rated games too...
I think you made a great decision by playing for 2nd here. This does illustrate my experience of playing for 2nd being a viable backup strategy in the three-player stage.