Does Biden actually want Ukraine to defeat Russia?

It's a big question which must have been discussed within the White House National Security Council many times. First, is it feasible for Ukraine to win a total victory against the Russian invasion force, driving them out of Ukraine AND out of Crimea, returning every inch of territory back to Kyiv? Realistically this seems a remote possibility. With Russia's firepower, never-ending troop reserves and Putin's determination to conquer or destroy Ukraine, such a victory would take years of fighting and there would be nothing left. A Pyrrhic victory. But what if Ukraine, backed by longer-range Nato weapons, manages to make life so difficult for the Russian troops in Ukraine, and adds to Putin's woes by dramatically stepping up attacks on targets inside Russia, that victory of sorts can be claimed by Kyiv. Would Putin be forced to backtrack in some way? This is the moment when the question in the headline comes to the fore. Would Putin become more dangerous if facing "defeat" and would the US and Europe suffer long-term as a result? Some might argue, wouldn't it be better, surely, for there to be a negotiated peace when both sides can claim something? And if this were to happen, could Russia/Putin be returned to the international family of nations and participate in global issues once again? Russia back in the G20 etc. None of these questions have satisfactory answers because the other argument is that Russia and Putin should be punished for attacking Ukraine and fully held to account. Putin in absentia found guilty of war crimes? That's what Ukraine wants but if it were to happen there would not be any hope of reconciliation between Moscow and the western world, and that would have high risks. What seems like a long time ago there was talk in Washington and in Nato about the need to give Putin a way out. In other words, some way of ending the war without totally humiliating the Russian leader. It's all too late for that now. How could it be morally right to let the man who is destroying Ukraine's civilian infrastructure to be allowed to take a prize home and carry on as if nothing has happened? So perhaps the answer really is for Russia to be defeated in this war. In which case the US and Nato have no choice but to give Ukraine everything they need to win this victory: tanks, fighter aircraft, long-range missiles and rockets. That would be war on a different scale. But how would Putin react to that? That's the question to which even the brightest members of Biden's national security team cannot have a definitive answer. So at least for the next six months, the war in Ukraine is going to be about stalemate and drift and uncertainty until Biden and Nato decide whether to go all-out for Russia's defeat or continue to hold back, as they are right now.