Why should US presidents be US born?

Companies look for CEOs outside their current workforce, universities perform nation or worldwide search for higher administration. Why do US presidents need to be born in the United States? Why restrict choices?

If you want the best person for the job, why would you but an artificial limit? For example, what is wrong with having an immigrant as president? The last two governor-generals of Canada were immigrants, the current one being even a dual citizen. Why does it even need to be a US citizen? If somebody proved to be a great president in an other country, why not move up to the United States? This certainly happens in the corporate world. And it even happens in politics: both Robert F. Kennedy and Hillary Clinton had no previous ties with New York before becoming its senators.

One argument I hear is that presidents are trusted with state secrets. Why would just the fact that someone is born in the country matter? As long as people are willing to vote for this person, citizenship at birth, or citizenship at election for that matter, should be irrelevant. A few examples: Sonia Gandhi, Italian-born, declined the post of prime-minister of India in 2004, several Canadian prime-ministers were immigrants, and there may be others.

And no, this post is not about Arnold Schwarzenegger in particular. I just fail to see the logic of this constraint.

PS: As this blog is apparently for a genious, here is something to lighten up the mood: How some Chinese children learn English.