The clause was actually carefully drafted to exclude Alexander Hamilton from the office. Anyone else excluded was collateral damage.
Hamilton would have been exempted by the same part after the first comma that exempted all the rest of the first Presidents. He would have been a citizen at the time of....
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No Person except a natural born Citizen, or a Citizen of the United States at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
Check your sources. Columbia University doesn’t seem to agree with you.
http://emoglen.law.columbia.edu/twiki/bin/view/AmLegalHist/WillHamiltonProject
During the period when a presidential run by Hamilton would have been practically possible, ranging from his resignation as the Secretary of the Treasury in 1796 to his death in 1804, a hypothetical candidate Hamilton would have easily met the latter two of the required criteria. In 1796, when his resignation as Secretary of the Treasury would have allowed him to run for President, he would have been 41 years old. Likewise, he had been a resident of New York for some 22 years in 1796, dating from his initial attendance at King’s College (Now Columbia University) since 1774.