They did set it higher. The President is the only official that must be born a citizen as opposed to naturalized.
What a relief! Except. . .
For the record, the Vice President must meet the same eligibility requirements as the president.
Chester A. Arthur was elected Vice President with a father who was a Northern Irish Citizen of the British Empire. William Arthur did become a naturalized US citizen when his son was fourteen years old.
Chester A. Arthur became president when James Garfield was assassinated.