The problem is, that Article II, Section I, Claus 5 presents a list of "qualifications" for the office of President. Amendment 22 was written with Article II, Amendment 12, and Amendment 20 in mind.
The proviso in Amendment 12, in the last sentence, was designed as a clarification. The Vice President had to meet the same qualifications as the President, rather than those outlined for a Senator. Prior to Amendment 12, the Vice President was the second-place finisher in the Electoral College. The Framers of the Constitution had not anticipated the rise of political parties. Now that the office of Vice President was to be voted on itself (in the Electoral College), the qualifications had to be clearly defined. And so they were ... the same as the President.