We must be careful “what we ask for, for we may surely get it”.
The overthrow of Putin will not end the war in Ukraine, and may worsen it. Putin, as the leader of Russia, is also a reflection of the will of the Russian people. A significant majority of Russians still support Putin and the war. There is a long way to go before the Russian people will be ready to stop the war or relinquish the territories already conquered. 2023 or beyond.
First, of course, Putin may become the symbol of Russian failure in Ukraine. Do not expect a groundswell of feeling that the struggle in Ukraine is unjust; that will be asking too much of the average Russian. The easiest target before turning on Putin will be Defense Minister Sergei Shoigu, who, by all accounts, has not done a very good job of the invasion and has never served in the Russian (or Soviet before) military.
It is easier to lose faith in one individual than in a national philosophy of victimhood and rejection. A facesaving narrative is difficult to reject, like the post-Soviet states joined NATO not because they were afraid that Russia would revert to its previous imperial dreams, but because NATO forced them to join and occupied the former Warsaw Pact countries.
As long as that kind of thinking remains the default message, unchallenged, there will be strife and a willingness to invade other countries. The rebuilding of the "legitimate Russian Empire" will not disappear as a national goal. There will remain resentment against all countries that joined NATO (they were too weak under pressure, and too weak to stand up for Russia) and against any country that considers joining NATO (cowards and traitors to their Russian people). The loyalty and obligation of Russia to Russians outside the Russian Federation will remain.
The Soviet Union was not a conglomeration of independent countries joined in a “union”; it was the Russian Empire under another name. The fact that some leaders came from constituent “Republics” does not change the very Russian nature of the USSR. Therefore, the fall of the USSR in 1991 was the second fall of the Russian Empire in a century.
If, and it is a huge “if”, Putin was overthrown (or were to die of a “natural cause”), we should not expect any new leadership to disavow the invasion of Ukraine, nor de-annexe the newly annexed territories. Crimea will not be on the bargaining table even if new leadership does agree to talk.
Personally, I discount any reports of Putin's health. He is as healthy as any national leader, but that requires a digression that I'll avoid.
Therefore, if suddenly he is no longer the President of Russia, it will have been through a coup, and a coup will not be staged by defeatists and the weak. Coups are not easy things, and they are rarely clean. Those perpetrating coups are ready for blood to be spilt, and the blood of their own compatriots who oppose their taking power. These are not the people who will seek to stop a war so that adversaries will no longer be hurt or killed.
Yes, hundreds of thousands are fleeing the country to avoid conscription in a poorly performing military. But these hundreds of thousands didn't try to leave before the mobilisation and threat of conscription. That is not the same as being willing to go out into the streets in adequate numbers to force a change. That won't happen, yet. More importantly, most Russians still support the war, though that support is eroding.
It will take time and more battlefield defeats to swing opinion definitively against the war and, when that happens, against Putin's leadership.
But National Honour will demand a victory, until it becomes very clear that such a victory is impossible. While we in the west may have come to the conclusion that a Russian victory is impossible, Russian media most certainly has not come to that conclusion.
Overthrowing, or otherwise replacing Putin (after he replaces DM Shoigu and keeps replacing generals), might not result in a peace-seeking government. It may result in another hardliner taking power and redoubling efforts in Ukraine. The real danger is that a replacement might decide to blur the lines with NATO and test NATO's actual resolve, potentially by hitting supply or logistics centres on the “other side” of the Ukrainian border with Poland, for example. As NATO definitely does not want to get into a war that may escalate into a nuclear conflict, a new strongman in the Kremlin may decide that NATO's redlines can be ignored.
A Putin replacement will know that he (and I am assuming that it will be a “he”) knows that he must deliver battlefield victories, quickly, or suffer potentially the same fate as the Kerensky government of 1917. Wikipedia sums it up best: " Despite mass opposition to the war, Kerensky chose to continue Russia's participation. His government cracked down on anti-war sentiment and dissent in 1917, which made his administration even more unpopular. Kerensky remained in power until the October Revolution."
Therefore, we should be careful what we wish for. And Russia should be particularly careful what it wishes for in the coming weeks and months.