No matter how many people you think there will be at the famous thing you traveled across the globe for, double the number twice. Your one-of-a-kind plan to be there at the perceived “down” time will end the same as someone who visits a popular thing on a whim.
Dealing with crowds is just part of the experience and you have to find a way to accept them as a fact. When a country has a feature worth visiting, people will travel absurd distances to see it. A great case in point is right here in the United States. We have 63 national parks that rack up millions of visitors every year collectively and these visitors come from all over the world.
You can just bet that no matter when you arrive, there will be a crowd, and it is best to reframe how you think about this situation. You get to share this experience with other human beings, potentially from a place radically different than where you come from. It is an opportunity to learn how citizens from around the world engage with the sights offered by countries that are not their homeland. It can give you a greater appreciation for how to best attend the places of interest in your own country.
I have found it is best to visit the popular places prepared to wait your turn to witness them firsthand. Be polite and allow others their moment to the best of your ability. It’s a fantastic way to build your patience. The world doesn’t owe you anything, and this includes preferred treatment at sights of significance.