The problem with looking at numbers, when it comes to the scale and casualty figures of a war, is that a) by the time they start shooting up very fast that's a lot of dead people that you can't ever bring back and b) the tenor and tone of a war are important in telling you what the war might do. The rhetoric from Azerbaijan, combined with its consistent attempts at heritage destruction and cultural erasure, are very obviously genocidal; some of the reports of mutilation of Armenian captives by the Azerbaijan regular forces are stomach-churning. If this war turns into a sizeable Azeri invasion of Syunik, and there is every reason to think that it might, then there will be every potential for full-scale civilian massacres. Also, the more one can make it clear that "small" inter-state wars will not be tolerated by the international community, the less other people are likely to start bigger ones. Setting expectations is important in diplomacy.
Another point, and one I don't think you've factored in, is that cost-effectiveness of action on a utilitarian level isn't just about the scale of a war. You can do a lot more to prevent deaths than you can to wade into a "hot" war where massacres are already taking place, the lines are a mess and you don't have a mechanism to pull anyone back. Conversely, you could probably stop Azerbaijan invading Armenia proper simply by signing a security agreement with Pashinyan that sat a token number of western troops along the Syunik borderline. Aliyev is a tub thumping nationalist bastard, but he's not a sufficiently isolated maniac to start a war that he thinks he might lose, so sending the physical message could well be enough to save a lot of lives if western countries actually want that outcome. I'm not convinced there's any military action we could take in Yemen or Tigray that would have similar cost-benefit analyses, and both of those wars are civil wars which makes the international legal situation considerably more complex: they're horrifying things, I think we should be spending vastly more on aid in those cases and putting much more pressure on governments and regional powers in both cases to try and stop the fighting... but when the context rather than just the numbers are taken into account, I don't think it makes any sense to relegate the current inter-state conflicts of central Asia and the Caucasus to some kind of "Tier 2" on the grounds that not enough people have been killed in them yet.
