I think there are three big factors that make involvement almost guaranteed. The first is that in modern times political leaders prefer to believe what they want to believe, instead of what is genuinely most probable. They won't see it as trading their existence for anything. They'll convince themselves that they'll decisively win, or that Russia won't retaliate, or won't be able to retaliate.
The second is that the leaders of these countries are very unpopular. Starmer has an approval rating of 22% which is somehow twice as high as that of Macron who has reached a simply impressive 11% approval rating. Politicians love nothing more than war when they're unpopular, because it gives the electorate something to focus on outside of their own internal problems -- Diversionary War Theory. Also I certainly disagree regarding our insecure 'peace president' who has an affinity for bombs and a trend towards megalomania.
And finally there is the wargaming results. Some of the actions that led to nuclear holocaust were relatively innocuous, including performative nuclear strikes intended to 'send a message' rather than actually cause much real damage. It doesn't even sound like a bad idea, at first. Make it clear that you're serious and this is the path we're going down.
But it fails to consider the most probable response. You want to send me a message? Okay, here's your answer. How do you like my message? And of course you can't let such a provocation then go unmet. It's easy to see how what sounds like a reasonable idea is in reality a very bad one, but modern political leaders in the West make endless poor decisions with predictably poor results. You've gotta be something special to have an approval rating of 11%. Were such things measured in the past, even Caligula would probably be looking down at you!