The question of who gets to be Tsar is an issue, since as has been brought up, the position of titular Tsar is disputed between two branches of the family.
The economic chaos of the 90s in Russia would probably mean the Communist Party under Gennady Zyuganov wins the elections and deposes the Tsar and his government (probably democratically). But then again, IOTL the CPRF and Zyuganov almost certainly defeated Yeltsin in 1996 yet officially Yeltsin won, so it's possible the neo-Tsarists can remain in power.
Or Putin could restore the Tsar and change the Russian political system so he can be appointed prime minister "at his majesty's pleasure" as many times as he can win elections. That would be far more legitimate than crowning himself Tsar as well as help legitimise his endless term in power.
The economic chaos of the 90s in Russia would probably mean the Communist Party under Gennady Zyuganov wins the elections and deposes the Tsar and his government (probably democratically). But then again, IOTL the CPRF and Zyuganov almost certainly defeated Yeltsin in 1996 yet officially Yeltsin won, so it's possible the neo-Tsarists can remain in power.
Who is the next heir? The family scattered to the winds. So finding a legitimate heir would be a problem. Then installing him without protest. Then there is the Russian Mafia and the post-Yelsin cronies that would want their cut. It would be as disaster in the making. Making Putin, Tsar of Russia would be far easier job since he is already there, has the power to be a Tsar, he has the army, and he does have most of the crown jewels in his possession - all he needs is a priest to crown him.
Or Putin could restore the Tsar and change the Russian political system so he can be appointed prime minister "at his majesty's pleasure" as many times as he can win elections. That would be far more legitimate than crowning himself Tsar as well as help legitimise his endless term in power.