Well, it is useful for the state to look bo worse than its population’s bad behavior or foreign nations’ bad behavior. Hitler was propped up as an icebreaker that way by communists willing to risk be sent to camps.
The Third Rome is a strange ambitious concept for Moscow aiming not at the West but at the seats of Constantinople and to displace Rome. I do not see much friendship from Moscow there.
The concept of the Third Rome is an invention of someone -- off-hand I forget the name -- in 16 century Russia. It assumes that Rome has already fallen, -- physically to the Germanic tribes and theologically to what the East perceives as error of Catholicism, and further to the heresy of Protestantism. The Holy See of the True Church then becomes Constantinople, but that, too, fell to the Turks; it is evident from this (quite prejudiced) observation that Moscow is indeed the Third Rome: the only standing Orthodox empire.
Naturally, Russia historically looked at Constantinople as its historical mission. I don't think any of that is relevant after a hundred years of disintegration in Russia.