He didn't really flee to russia, he was on his way to Ecuador, when the US cancelled his traveling documents trapping him in Russia because it's way easier to discredit him while he is residing in Russia
I think technically the problem wasn't that the US had cancelled his travel documents, as the Ecuadorian government could grant the necessary document to a refugee (which is almost what happened[0]), but rather that he was worried the US would force his plane to land in a country where he could be arrested.
That fear was well-founded, given what later happened to Bolivian president Evo Morales when the US suspected he might be transporting Snowden out of Russia[1], even though forcing planes to land to capture political prisoners is more fitting for a dictatorship than a democracy[2], but I suspect that Russia made it a condition of Snowden's safety that he not attempt to travel to any other country.
Of course I'm not saying I would have done anything differently were I in Snowden's position, but it is worth noting that he could in principle have avoided US and NATO airspace by travelling from Russia to Shenzhen to Johannesburg[3] to São Paulo[4] to Ecuador.
