I dont see how KC and Monument could every lead to a stalemate. First of all, unless you trashed with Raze, there is a sixth card in your deck which prevents a flat 9VP per turn. Second, building an engine around that combo seems like the natural way to go.
It is highly unlikely that even in a mirror, players will arrive at the same deck situation symmetrically and then whoever was faster has an incentive to, well, yeah actually win the game.
Banish, Butterfly, Bonfire ... and that is just the Bs, there are a lot of different ways to get to Kc/Monument. There are many ways to get a trasher out of your deck.
Building an engine requires some form of +cards. If you lack that, then the max per turn is 9 VP. And even if you have it, there is no assurance that greening is not a zugzwang move (e.g. Hireling can let you build out to 12 VP/turn when you play them, but at the risk of slowing your VP gain with bad shuffle luck).
Which is where the stalemate comes in. Even with a dead trasher, 6 VP/turn is dominant over going for greens with only one turn per hand. Big money monument runs will be hard pressed to score 7 VP all that many turns and even if you alternate with 4 VP using duchy, the 6 VP/turn golden deck wins out. If you have Kc/Mon/Banish, forget it, 9VP/turn will overtake provinces very shortly and once you have the setup, there is significant downside to buying more Kc, Mons, let alone something else.
Ending the game requires there to be piles that you can safely buy out before the opponent running the combo overtakes you. Which is not exactly easy. After all, say you get there 3 turns faster. Call it a 15 VP lead. How many turns does it take to pile out? Well, assuming 10 of 20 cards are bought for Kc/Mon, that only leaves you 20 more turns to pile down via three pile. Ending the game then requires you to average <1 VP lost per turn to shuffle concerns.
What stops this from showing up every time is:
1. Boards often have engine potential.
2. People often do not see utterly dominant combos
3. People tend to resign even when they are stalemated if they are unable to win.
Boards without engines are less common, but Stampede/Collections stalemates are not categorically different than something like Kc/Monument/Banish.
It's not a weird fascination, it's the established standard way to play.
Established by whom?
It is the most popular way to play and the norm for competitions ... but on the former category the overwhelming response I get when I play a stalemating combo is for people to just resign after the first few cycles as my lead grows. On the latter it seems rather trivial to adopt some tournament rule to break the stalemate (e.g. a player who maintains a lead for 50 turns wins the game, with P2 leading in the event of ties) or to prevent whichever combos are deemed too likely to stalemate (e.g. shuffle up a new kingdom if Collections/Stamped comes up, maybe for Stampede/Ride and Stampede/Supplies absent engine enablers).
I mean I could be wrong, and maybe somehow ultracompetitive play with Possession does not become a degenerate mess where you want to instantly tank your own deck and just mass Possess the other guy's deck, but I would suspect that Possession has far, far more stalemates than some once-in-blue-moon two card combo.
Holger:
Any two card combo is more common than any particular 3 card combo, but there are almost always more of the 3 card versions and the question is if there are enough more of the 3-card variety to nonetheless be the dominant possible stalemate trap.
And I am not convinced that perfect play would not find more of the three or more varieties.
After all, those are hard to spot. People here are having trouble even being exhaustive about how to setup a simple 9 VP Monument deck. Yet it can be done with both Kc and Mm (though the latter cannot hit 9 VP/turn in a perfect mirror), but you can also manage similar shots with things like Tr/Mon/Outpost or Tr/Mon/Mission, and of course, Tr can be replaced by Rg or Rc in some circumstances. Even when trying to be exhaustive about something as simple ditching a trasher, people are missing things like Necromancer who can trash the trasher and 3 Mon to setup the 9 VP/turn option.
End of the day, the point is that we have had a LOT of very powerful things that went unnoticed for months or years. Masq pins (pre-errata) were possible from day 1 for Seaside (Minion/Tr/Tr/Outpost/Masq), yet it was years before somebody put the pieces together for Goons/Kc/Masq. Have we found all the stalemates yet? I doubt it. We may well find out that at skilled enough play (maybe some sort of alpha-star-dominion), stalemate is the normal endgame as both players reach a point dancing where buying the Nth kingdom card lowers the odds of winning more than passing and letting the opponent buy it.
Which is my main contention here. The simple errata is to remove Horses from token gaining. The simple solution to stalemates, in general, is just to adopt a 50 turn rule where anyone who leads for 50 rounds (P2 "leading" during ties), wins the game.
Stampede/collections is "solved" by either and nowhere near unique in its function.
Segura:
Even adding in draw does not negate the stalemate. You need two Kc to start the chain absent +actions. This means that roughly 40% of your deck needs to be Kcs to have decent odds of hitting off a chain. That means your deck needs to be <12.5 cards. Maximal VP gain is 15 VP/turn. Kc and Mon take up 10 slots and leave you with ~3 for draw cards before your expected VP total drops. If your opponent goes for 9 VP/turn and you opt for engine, your game ending deck is 7 Kc, 7 Mon, and 10 Draw. Median hand is ~5 VP/turn at that point. And absent additional gains, that means you played for 19 turns after a mirror setup absent additional gains.
Maybe these things are rare on all random, but I would submit that playing them more often gives me a much better feel for how a particular set actually plays to stalemate.
What makes the engine more viable is (+action and +cards) or (+cards and +buy/gains).
And again, it is not like 6 VP/turn is precisely bad on a board with no +buy. It generally wins out against the province piler absent enablers as 7 greens makes for a lot of 5VP losses per round.