There are no such assumptions. If Russia shoots US troops that are simply defending themselves while trying to keep a Russian army from advancing into Eastern Ukraine, the US will probably to go to war scale war with Russia.  It's that simple.  It doesn't matter what Russia claims because it is the US population and political class that makes that call, not international opinion.  Same has been the case with Putin's risking of Russian forces.  Full on war with the US means the game would be over for Putin, since Putin doesn't want a war, and neither does anybody else.  Putin knows this, but he has been betting that the West won't take the same risks that he is doing, largely because he knows that Ukraine really isn't very important, resource or strategic-wise, to the US or Western Europe.

The chess match that Putin has been playing involves putting his forces in places that risk nuclear war or full-on conventional war to remove them.  Like the classic prisoner's dilemma game, it means only a willingness to do the same on the part of the West is capable of thwarting Putin's strategy, a condition which is inherently unstable because of the uncertainty involved and so unlikely to be made by the West.

If the Ukrainians were to react negatively to US troops entering their country, which is always a possibility, then such a strategy to match Putin's risk-taking won't work, I agree.  It is dependent upon support from Ukrainians.  My sense right now, however, having talked to many different Ukrainians over the last few weeks, Russian as well as Ukrainian speaking, is that such support would be forthcoming for the US at the present time.  They really don't want Russia coming any further right now.

The West, meaning the US all alone, certainly can "win" a conventional war against Russia in the Ukraine. Such an outcome is not guaranteed, but both Russian and US generals know that this is the case. A conventional war in Eastern Europe is actually the war that the US is best equipped to fight as well, and it is the one that it has spent more resources, time, and preparation training for than any other, Afghanistan and Iraq notwithstanding.  The US is still equipped mostly with pre-2000 era heavy equipment exactly designed for that very fight, so although that scenario is also the one Russia is best equipped for, it is still the one NATO has always had all the equipment already in place in Europe to do as well. That's what makes this crisis such a scary prospect, similar to the conditions at the beginning of WWI.

However, if a war of any kind starts, it's a loss for everyone, because it really would be a big one, and both sides know this and are specifically trying to avoid it while bluffing each other and making strategic moves that require the other side to back away when war is the only other option. So far Putin has outplayed NATO in every way in this chess game, largely because he has been willing to take risks the West has not.  He has been willing to play chicken with the prospect of full on war with the West, which means that unless the West is willing to do the same, it cannot possibly gain an equivalent standing in any negotiation.  

Not being willing to do that means allowing Putin to have the upper hand in negotiations.  To me, that's perfectly okay and is probably the best thing for the West to do in this case, precisely because the Ukraine really isn't that important to anyone except Russia, and really should never have been staked out as a NATO objective in the first place.