The US has long recognized that Putin is a thug, and that Russia's security would be threatened if Ukraine were allowed to join NATO. The US has, however, refused to consider this same security externality when it comes to Russia's aggressions towards Ukraine.
The Ukraine conflict could have been prevented by engaging in diplomacy, and the deal that could end the war is for Ukraine to remain a neutral state. The question is, what is Putin willing to do in terms of evading other countries?
The decision by George W. Bush to expand NATO by adding Ukraine or Georgia would not enhance the security of the US because it would threaten the relationship between US and Russia due to Putin's nationalism, which is the source of his popularity in Russia.
The annexation of Crimea by Russia came after Ukraine sought an association agreement with the European Union, not with NATO at that particular point, leading to Vladimir Putin citing security concerns about the expansion of NATO as a reason to invade.
The use of a pocket nuke by Russia could bring a specific benefit to them, as they act without bureaucratic gridlock, unlike democracies. The United States could pressure NATO to come to a certain conclusion, however, NATO may be hesitant to act out of fear of being bullied by the US.
The US needs to build an alliance with Pakistan, India, and South Korea in case of China invading Taiwan. It may be possible for the US to make a deal with China, focusing on the negative impact of missiles in their own backyard.