Actually the Constitutional language is "eligible *to* the office". I would think that would include running for it. But then again, no one "ran" for the office in the early days, and the electors were better known in their states than the candidates in some cases. And people voted for the electors, not for the candidates.
In any event, we need laws, at both the state and national level, requiring candidates for federal office to document their eligibility for the office for which they are running. Why would/should the country have it's time wasted by a candidate not eligible to the office?
to : 4d Chiefly Archaic. To serve as; in the capacity or part of; as, to take one to witness. "Water to his drink." Chaucer. "Wilt thou have this man to thy wedded husband?" Book of Common Prayer