The problem with this is Russia could launch missile strikes without putting a single troop on the ground. How do you repel that just to your borders?
I don't see what sending troops across a border does to address something like that. First of all, that's a different matter from engaging troops who do cross into NATO or surging across Ukraine, and then into Russian territory in either case. You're talking about something of a different order. If these are conventional missiles, every dog gets his one bite. If the Russians say, "Whoops, my bad," then you make it plain that that was their one "mistake", and any others will result in similar action. If it's nuclear, well, then expect to evaporate by the end of that day, because the escalations will come fast and furious after that. So just pray that never happens.
Secondly, aside from Kaliningrad, we're hundreds and hundreds of miles away from any frontier with Russia itself, other than the rather tricky access we'd have in the Baltics, and Russia's pretty likely to roll in there mighty fast if this becomes a shooting war. So almost any troops we send into "Russia" are actually going to be met in either Belarus or Ukraine anyway, not Russia proper.
You have to, at the very least, take out military assets in Russia proper. You can of course do this without an invasion force of your own, but certainly missles would rain down on Russia in the event of an attack on NATO. There is no way around that.
You make it sound like a video game. What you're talking about is very likely the end of the world. You're talking about sending missiles into the country with the world's largest nuclear arsenal, and very little else, at this point. You're talking about destroying infrastructure on Russian soil and killing Russian citizens. Do you think they're just going to sit back and say, "Ah, you got us, guess we're even now"? They won't. They're going to fire back. And then we're going to fire back. And then they're going to obliterate one of our cities. And we're going to obliterate one of theirs. And at that point, it's very unlikely that we, any of us, will be able to put a stop to a general nuclear exchange. Nobody gets any points on a screen out of this. Nobody lives to compare scores at the end. Everything we've done for the past 15,000—hell, 2 million—years ends
. So all I can tell you is, you'd really better pray that first missile across that frontier never gets fired, because that d!ck-measuring contest ends like Reservoir Dogs.