It's not "foreign-born" per se who can't be Prez.
Foreign-born Americans can already run (Goldwater was born in Arizona before it was part of the union, for instance). Incidentally, the best president who never was president was Alexander Hamilton, who was not able to run, having been born (illegitimately, no less) to a British ex-patriot living in Jamaica.
American-born children of foreign parents are also able to run as they are automatically citizens "through geography" as they say (though an American-born child of foreign-born parents who were never naturalized can actually be deported under some circumstances).
So it's really just foreign-born children of non-Americans who are barred from the Presidency. This could have theoretically created a problem when Kissinger and Brezinski were Secretaries of State in their days, as neither was born in the US and therefore would have had to have been skipped in the order of succession (but if they ever have to go all the way down to the Secretary of State, the US Government would probably have already been taken over by FEMA).
Barring any new Alexander Hamiltons, there are just two words as to why no "foreign-born child of non-American parents" should be allowed to run for President: GEORGE SOROS
Thanks for the information, it's quite interesting. Let's just say, we don't need to amend the constitution on this point :)