Sorry, but this is bad history.

Russia never called itself the Third Rome. The city of Moscow was called the Third Rome and by a few church figures sucking up to the Tsar in the 15th/16th centuries after the Ottomans took Constantinople, but despite this Russia acknowledged the Patriarch of Constantinople (then a vassal of the Ottoman Sultan) as the leader above the Patriarch of Moscow, though its church acquired autocephaly (autonomy).

Then some scholars in the 19th century revived the idea while Russia was already expanding at the expense of the Ottomans and trying to claim the right to defend the interests of the Orthodox Christians of the Ottoman Empire, but it remained an idea in literature, not a literal title or diplomatic claim.

So, as with much in history - it's the reverse of how it looks on the surface: the ideological follows the material. Russia wasn't trying to take Constantinople because of some medieval claim, but rather it had been gobbling up Ottoman possessions for centuries, the strategically located Constantinople being a chief prize, and then sought to legitimize this process with some scholarly discourse after the fact.