I think you're giving too much credit to the US here and too little agency to the Russians.
Let's be very clear here: Russia had a choice. I thought up until about a week ago that this was simply then playing hardball on the Minsk agreement - a framework agreed upon by Russia, the US, the EU and Ukraine that in theory would have resolved Russia's concerns about NATO on their borders without dismembering Ukraine entirely. The essence of it would have been that the separatist regions of Ukraine would have been recognised as having a special regional status and greater freedoms to institute their own laws but in return would get a veto over future national security arrangements. This would have allowed for Russia to covertly control them from behind the scenes and use them to keep them out of the NATO framework. Again, this was agreed to by the US, though Ukraine was dragging it's feet on implementing it.
However, we can now clearly see that was not the case. Indeed, Russia was engaging in duplicitous diplomacy with France and Germany right up until the point the invasion started precisely to convince them that this was their aim.
Furthermore, the Ukraine of today is not the Ukraine of ten years ago. Despite not being a member of NATO, it's armed forces have been considerably hardened by US and EU aid, most notably advanced Stingers and MANPADs were unlocked for sale a couple of months back - in conjunction with everything else they've been given over the past eight years, they have the means to turn Ukraine into a hellish insurgent landscape. The kind of urban fighting that commanders hate and irregular fighters love - that's what awaits the Russians currently in Kiev and Odessa, and they are going to be bloodied night and day until they leave. Bombs on the street, rat poison and glass in their food...it's never going to end and they'll be looking over their backs every moment they're there.
Finally, Russia's economy cannot afford a protracted conflict, and nor can their military. They are trying to run a superpower on an economy the size of Texas. Three-quarters of their available manpower is now concentrated on Ukraine or the borders around it. That means they are weaker everywhere else - and the longer this goes on, the weaker they will get. NATO won't take direct advantage of this, because no-one wants two nuclear powers fighting - but you can bet Russia's partners in the Middle East, the Caucasians and in Central Asia will feel their absence.
In short, there were a lot of reasons to believe that Russia would not invade, because invading is about the dumbest thing Putin could do. But he did. Putin chose to wage a war of aggression, when he had other options available, and the reasons for that are complicated but essentially there is a revanchist, nationalistic movement within the Russian "mainstream" that wishes to rectify the "mistakes" of history, such as the dissolution of the Russian Empire and it's successor state in the Soviet Union.
This movement views countries like Ukraine and even Belarus as illegitimate creations of the Soviet state that should have returned to a Russian status at the end of the Cold War. It's this movement which managed to get a vote through the Russian Parliament that those Ukrainian regions be recognised as independent - and certainly it can be argued that such a proposal never would have made it through without being agreed on from higher up. But Russia is not a straightforward dictatorship where a single man rules - there are factions and key constituencies who need to be listened to and supported, and there is negotiation back and forth between these groups and various power centres in the Russian state, which includes oligarchs who stand to profit not only from a conflict in Ukraine but the establishment of new markets where sanctions are not applied to them. These power centres, for their various reasons, decided a Ukrainian invasion was the way to go, and so allowed the vote to go ahead.
That's not to say that the US and NATO do not share some blame - they certainly could have done more, both historically and in the present to try and assure Russia of it's security aims. At the same time, given what has already been provided, it's hard to say what would have actually convinced the Russians to back off, without a complete change in NATO policy going back to the early 1990s or similar. A democratic Ukraine was always going to be on contentious ground with an autocratic Russia - and would naturally seek allies and frameworks agreed on with them to try and blunt any Russian aggression. The lack of natural barriers - barring the Dnieper - in the region mean security is always going to be fraught and hard to obtain in any concrete way except through these alliances and agreements, and short of telling Ukraine to fend for itself and leaving it to the Russians to absorb, I think any degree of assistance was always going to be looked upon by a suspicious Kremlin as the first step in a NATO agreement. In short, Russia views Ukraine in a simple binary position of either it is with them, or it is against them. Clearly it is not with them for now, so the only thing to do is secure a regime change to ensure that is not the case in the future.