The World Government believes that the large Straw Hat is Uranus, one of the 3 Ancient Weapons
Over 800 years ago, the leader of the Ancient Kingdom was a giant who wore a Straw Hat. The people of the Ancient Kingdom were the original clan of D., and their leader who for simplicity’s sake will be called “Leader” was one of the early people who possessed the will of D.
He was the man most feared by the 20 Kingdoms because of a strange ability he possessed: He had the power to create an ally out of anyone, friend or foe, and amass an army in mere days. That power was called “Uranus” by the 20 Kingdoms, yet none of them knew why he possessed it or how it worked.
Knowing that they’d soon be defeated, the Ancient Kingdom decided to pass their history down to future generations via the Poneglyphs.
In addition to that, they decided they needed to hide their 3 Ancient Weapons. When it came to Uranus, the Leader devised a plan that would not only fool the World Government, but would at the same time shift their attention onto a useless hat made of straw, when the real threat and the real power of Uranus (the Leader himself) was killed instead. How he managed to fool them into believing that the large Straw Hat itself was Uranus is beyond me, but it isn’t important in this discussion. He might have fed it a Devil Fruit or performed an action on it that made it seem powerful, who knows.
As for the small Straw Hat that was newly made by him, he passed it down to a young kid from the D Clan and had him escape before their Kingdom’s annihilation. He was to travel the world and amass enough power to take those 20 kingdoms down, but as he reached an old age and realized that it just wasn’t possible, he passed his hat down to a young member of his crew along with his hopes and dreams. And just like that, the cycle began.
The important thing to take note of is that the World Government acquired that large Straw Hat and stored it in what would later become the Holy Land of Mary Geoise. For the longest time, they believed that they had Uranus, one of the 3 Ancient Weapons stored in their vault, and even though they didn’t know how to operate it, just the fact that they had possessed it was good enough, believing that there would come a day in which they were able to wield it.
So ironically, a large hat made of straw was passed down through generations within the World Government, each generation unable to make it work, while the real threat, the Will of D. symbolized by a smaller hat made of straw, was passed down through generations in the shadows, each generation growing stronger and closer to defeating the World Government.