But NATO doesn't really want to fight that war. Electorate would vote governments out for getting involved in war they doesn't really care about. And Putin knows that too.

So what if NATO puts some forces in Ukraine, but Putin calls their bluff and attacks anyway? Small contingent wouldn't be nearly enough to hold the territory, so you need to either engage for real (but you don't have political capital for that), or withdraw forces which would look like huge humiliation.