Not English Common law. English STATUTORY law. Get it right.
Between the adoption of the Constitution in 1787 and the end of the war of 1812 - in 1815 - Presidents Fillmore, Pierce, Buchanan, Lincoln and Johnson were all born.
Were none of these U.S. Presidents qualified because England considered them English according to English law?
If we had been respecting English Law, then the answer is "yes." Since we didn't respect English law, and had instead replaced it with American law, the answer is "no."
So you hold that the laws of a foreign nation has no bearing upon the natural born citizenship of an American?