You can go in and manually turn on “Capture the King” in settings to play this way.
Allowing king's suicidal moves in 4pc

there is a version of four player chess called "4pc max" (with capture the king enabled). You can even play rated games.
there's also a game rule called self-check which allows you to gain points off of mating yourself
Is there a game where the 2nd, the 3rd, and the 4th gain equally and the 1st loses?
Because I think I need that game.

that would be the opposite of solo but no, why would that exist? Then people would intentionally play for last place and it would be a completely luck-based game
like in Chaturaji, may make things more interesting, or at least different in a not uninteresting way.
Ex, if a king can be allowed to move right next to an enemy queen, because it's 4pc and the other two players can still make moves that prevent your opponent from eating your king and so your king can possibly eat an enemy queen. King then can be a more active piece. But I suppose there was some good reason not to do that in 4pc chess though I don't know what exactly.