That’s a pretty good way of putting it. Both the governments of Europe and North America have pretty well moved away from honoring or even respecting Christianity, so Russia could move into the vacuum, culturally positioning itself as a “defender of the faith” against those who would disparage it.
As a ‘passive’ move, requiring few resources, it could prove very advantageous to Russia in the future. Likewise, it could be very beneficial to Rome, currently surrounded by a hateful Europe, to have a powerful spiritual ally in Moscow.
Worth a try, seems like. But what do I know? Putin seems Machiavellian enough to do it.