Was the basilica of St. Peter built because Byzantine fell?

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So as you all know, Byzantine empire and Roman empire had a dispute over who was going to rule over christendom. This dispute lasted some hundred years, and the tensions got bigger for each century. After letters were exchanged between the ruler of Byzantine and the advisor of the Papacy, the Great Schism occurred, and now the church was officially divided in two.

Given that the dispute was about authority and power, wouldn't it be fair to conclude that the new St. Peter Basilica was built because of the fall of Byzantium? After the fall of Byzantium, the churches was converted to mosques, Hagia Sophia, the St. Peter's Basilica of Byzantium for instance. Eastern christendom had lost all its power. The fall of Byzantium is marked 1453.

Now, from the Pope's perspective, wouldn't it be a big strategic move to build a grand construction that will show the world where the capital of christendom lies? This is what the Papacy has wanted for so long; complete rule over christendom. The construction started in the year 1506. In my opinion, the construction of the Basilica of St. Peter, really was a power demonstration, and this isn't so far fetched to conclude when you look at how catholic policy was conducted in the upcoming years; striving for complete power, focusing on the materialistic, don't let scientists challenge their beliefs.

So guys, what do you think. Was the Basilica only a demonstration of power? I cant seem to find any sources on this. Makes complete sense to me.