Final answer:
Alexander Hamilton's eligibility to run for President wasn't questioned, but his political actions, such as opposing John Adams and involvement in a duel with Aaron Burr, shaped his historical significance without a presidential run.
Step-by-step explanation:
The reason Alexander Hamilton could not run for President is not directly stated in the provided reference material, and historically, it is clear that his eligibility to run was not in question. However, Hamilton's influence in the elections of his time, and his views on executive power, can provide context to his political maneuverings during those years.
In the Election of 1800, Hamilton strongly opposed John Adams within his own Federalist Party, even though they had endorsed Adams along with Charles Pinckney. Hamilton leaked a report criticizing Adams's abilities for the presidency, indicating "great defects" in Adams's character, despite their affiliation to the same party. Also, in Federalist No. 68, Hamilton argued why it was not dangerous to place power in the hands of one man, as electors from each state would select the President and be free of corruption and external influence.
Hamilton's active engagement in the politics of the time, particularly his rivalry with Aaron Burr, whom he disparaged and ultimately engaged in a fatal duel with, shaped his legacy more than any run for the presidency itself. As a key Founding Father and creator of many foundational financial systems, Hamilton's life ended too early for him to make a bid for the presidency.