Although Putin is a despot, he is still only one piece in this chess match of the global competition. His options and decisions are constrained by others. On the other hand, Biden has the greatest influence and most options to dictate what will happen next.
At the moment, he decided to push for a Russian defeat by escalating the military supports to Ukraine because Ukrainians were clearly willing to pay the price for a total military victory against Russia. The fault in this path of action is really an assumption that this conflict will not escalate to a nuclear Armageddon.
The question that needs to asked and answered is what will be the US reaction if Putin uses a tactical nuclear weapon in Ukraine. Here are possible options: 1. Direct NATO/US involvement in the war and they will try to fight it with only conventional weapons, which US can certainly win; 2. a limited nuclear strike on Russia as a warning; 3. all out strikes against Russia with nuclear strategic weapons. The sad truth is the option 1 and 2 are not really viable options because Russia will not be willing to be constrained to a type of war that they will lose. So it will be a road to Armageddon.
The next question is whether Putin will push the nuclear button when Ukrainians defeat the Russians on the battlefield by reclaiming eastern Donbass and Crimea. Biden's advisers seem to think that Putin will not. Plenty of people think that Putin will. It is interesting that there may be a correlation between the perception of Russian strength and Putin's willingness to use nuclear weapons. It seems that people who perceive Russia as strong believe that Russia will use all means to fight (i.e. 5-6k nuclear warheads) whereas people who think that Russia is militarily weak have taken nuclear weapons out of the equation. This seems to a peculiar mental blind spot for neoliberals in Biden's administration. Is Putin a modern day Hitler or not? If he is, why would he refrain from going nuclear? If he is not, why the war must continue at all costs?
I think that the smarter option for Biden is to declare a victory, which is a real one by the simple fact that Putin failed to topple the Ukraine government. A ceasefire at the existing line of contact with maintaining economic isolation of Russia should be viewed as a great outcome from the US perspective. Pushing for more would mean a real and substantial risk of nuclear Armageddon.