I think the question is: how successful is Russia going to be with its alternate Olympics? What countries are friends with Russia and so would even be open to such a thing?
North Korea and China, for sure.
Belarus? They are banned now but the sporting world has distanced itself from Russia's invasion of Ukraine and have tried to position its athletes as not only neutral but also victims of the invasion. Joining in with these "Friendship" games would kill any chances they have to get back into their sports at the world level.
Iran? India? Syria? They are actively supporting Russia in the war, as actively as NATO countries are supporting Ukraine. They also aren't a force in many Olympic sports, especially winter sports.
A handful of tiny countries like Myanmar? They have supported Russia's invasion with words but will they turn their back on the Olympic movement to please Russia?
Theoretically, there is The Collective Security Treaty Organization that includes Armenia, Kazakhstan, Russia, Belarus, Kyrgyzstan, and Tajikistan. But not all of them have supported Russia's invasion of Ukraine and some have already made public statements distancing themselves from Russia.
Even if every single one of these countries were to sign on, it doesn't seem to me to be enough to have a real competition that would rival the Olympics. And I am doubtful that all of them will sign on. Outside of these countries, no NATO country is going to sign on. I guess there are countries that right now are neutral (not in NATO, not in the CSTO, not publicly an ally of Russia) that might sign on but what would their incentive be?
I am actually all for a better alternative to the IOC who I believe have corrupted the games too much. But this ain't it. It's more like rival mafias fighting for territory.