Why did he have to say anything? As you said, he's only responsible for his own organisation.
He doesn't have to
say anything, but I think it's just diplomacy. He's responsible for his organisation as you say, but his organisation operates in countries (and has for centuries in some cases) where those religions are a) the official religion, b) the religion of the majority, or c) influential in the country. Also, in some of these countries, the RC's (and Protestants, to an extent) are viewed as problematic not just because they're Christians but because their highest religious allegiance after God belongs to a foreign national (in fact, a foreign head of state when it comes to the Pope). Even if they are natives, they're also looked at as foreigners, a sort of Trojan horse within "our country". In parts of the world, their priests and religious are martyred for this allegiance, if not overtly for the faith itself.
It's understandable for the Pope of Rome to want to get ahead of all that and be friendly and diplomatic. Those messages don't typically affirm people in their paganism, even if they don't outright preach the gospel of Jesus Christ. They're usually a kind of PR fluff that's aimed at easing relations, helping others see the RCC as a friend and not a threat, etc. I don't know if any Orthodox leaders put out "official greetings" the way the Pope does, but there are plenty that will go to public functions on the occasion of holidays not to be a syncretist or a fundamentalist, but to be a friend and an actively engaged member of society.