Ooh, careful, Raf. Don’t poke the bear! 
Kidding aside, your point is valid. If fact, I think that the Presidency may well require greater stamina owing to the fact that it is a 24/7/365 position with little down time. However, I would also posit that age limits should be equally applied to Congress and the Supreme Court! Considering the number of fossils inhabiting the halls of Congress, the Capitol building could be labeled as an archeological dig site!
Again, kidding aside, many of the age limitation standards in use today were done at a time when the average lifespan was considerably shorter than it is today. They are also used, as other have said, because a set age is a simple and predictable end point that is equally applied to everyone in that profession. Setting an “on condition” limitation is probably much fairer, but it requires both the government and the medical profession to establish physical and mental ability rules that would be constantly contested by someone who was just rejected during an exam. Given the choice between easy and fair, the government will always choose the easy route.
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