JKCheeseChess take
I partially disagree. 4pc should seek to be as logically similar to 2pc as possible, for purely functional reasons. We do not have the playerbase etc. to survive on our own and so we need to seek to be as attractive as possible to 2pc players as possible.
For example, we have En Passant. This is arguably a very stupid rule (in regular chess, it is more of a headache to beginners than anything else) but the reason we have it is because regular chess does.
The point here is I think it is very important that things should line up to 2pc as possible. Not to the extent of making 4pc king capture rules nonsense (as would be a hypothetical rule that would make king capture impossible to preserve 2pc checkmating situations) but I think it should be a priority.
I simply disagree that "tactics built around double checks and discovered attacks" are crucial to 4pc. They are, but imo not in relation to the question. At minimum, looking at the game, I would not be frustrated in the least if the option to king capture were taken away; like En Passant it doesn't seem to matter much either way. Seeing as these king capture moves increase things at most one cycle in depth I don't think it's fundamental. I can't think of any other games off the top of my head but if everything is like the game mentioned in the beginning it doesn't seem to matter much; it's fairly arbitrary to the complexity and skill of the game
I guess my only argument here is that chess variants are variants for a reason. Everything you've said is factually correct but in the context of 4 player chess and other variants it is irrelevant, simply because the inherent mechanics for each game is different. For example, regicide is its own game rule, where every player is allowed to "hang their king" (this means checks, pins, etc. aren't forceful moves). Another example is stalemate: some variants are programmed to have stalemate be a condition for losing, which aligns with your "Chess is not about trapping the king; otherwise, a stalemate would also be a victory." statement that you made in response to my albeit flawed explanation of checkmate. In some games, stalemate is a win condition. It all depends on what we are talking about.
What I'm trying to say is that we can't simply change the rule for one specific case because that would break so many other cases where the mechanic is viable, maybe even crucial to the gameplay of that variant. Especially in 4pc, where many tactics are built around double checks and discovered attacks, removing this mechanic because that's how it works in a completely different game would be a silly idea.
That's my take