Man, come on. If I have a house rule about checkmate scenarios and I give you an example of one I consider to be a checkmate scenario, isn’t it a little bit unfair to argue that example doesn’t meet my criteria?
I don't think I argued that, but ok, now that you mention it, the rope bridge scenario may meet your criteria, but it in no way represents the same problem for D&D that "knife held to the throat" does.
As Umbran has helpfully pointed out, the thing that D&D cannot cover by its standard rules is a called shot and in particular a called shot to the throat since "the throat" has no meaning to the rules. Whereas, everything about the bridge is, as I've already pointed out, things covered by the rules. That makes the two things different, even if they aren't different under your checkmate rules.
It is not at all obvious that the rope bridge needs "checkmate" rules nor can I see how applying the "checkmate" rules solves a problem there, since the same outcome can be achieved without them. Regardless of whether your criteria can cover the rope bridge scenario, that is at the least a categorical difference between the two.
I mean, “here’s an example of the sort of thing I’m talking about” and you respond “no it’s not.”
Well, I mean, it's just not.
I suppose you could apply your "checkmate" rules to the bridge scene, but the point I'm trying to make is that to achieve the same outcome (everyone who falls without an answer to the fictional positioning of falling from a great height, dies) does not require special house rules, just a sufficiently deadly fall. D&D handles the rope bridge scenario just fine. There is one in X1 Isle of Dread, for example with a 2000' fall for like 5th level characters (surely lethal), and as D&D has acquired a more generic and universal skill system, the way to properly run the rope bridge scenario has become more and more standardized so that even the DCs and the sorts of fortune tests involved are specified explicitly by the rules.
Whereas, for the last at least 30 years everyone familiar with the game rules has encountered the "knife to the throat" scenario, been troubled by it, and often considered what sort of rules would need to be added to the game for the scene to play out intuitively. I remember talking to fellow players about this very same problem in like 1991, and there is still not to my knowledge a fully satisfying answer.
If you have some codified rules that you can apply to generate or arbitrate "checkmate" scenarios, I'm inclined to think that either they are entirely arbitrary or else that ignoring them will work better for the bridge case. That's because if the bridge ropes are easy to cut, then no elaborate contest to achieve "checkmate" is necessary to defeat enemies "automatically" on the bridge. Why in this case is "automatically" a needed thing? Declaring, "I cut the rope." is enough. In most games, "I hold an action to cut the rope." also works just fine with no special rules and no "checkmate" required. And certainly, if we did have such a contest on the bridge and want to make it interesting and high verisimilitude (something you seem to require in all this discussion), we'd have to take into account how far apart the rope cutter would be from whatever he's trying to "checkmate", what weapons and tools he had available to him, and what weapons and abilities his foes have, and all of that can be easily handled by D&D's normal process resolution. To handle all the factors that D&D's normal process resolution handles at least as gracefully as D&D's normal process resolution, if it "checkmate" was applied to the bridge scenario as well as the knife to the throat scenario, then your "checkmate" system would either have to be at least as complicated as the whole of D&D's normal process resolution, or else incorporate the whole of D&D's normal process resolution as a subsystem.
But, if it incorporates the whole of D&D's normal process resolution as a subsystem, and the subsystem itself can handle the bridge scenario, and then adding the checkmate system adds nothing to the bridge scenario.
So at this point, I can sum up my whole opinion of how this discussion has gone, by "Your bluffing. You have no cards in your hand." I call. Copy paste your "checkmate" house rules to the board for us to analyze.