Please find "being nominated for President" or "running for President" in the Constitution.
Eligibility to the office of President is a constraint on the Electoral College, and arguably on the Joint Special Session of Congress that counts the electoral votes, or the House of Representatives if it gets that far.
But any person, eligible or not, can "run for President" without creating a legal or Constitutional issue.
Filing a false certification of eligibility is a serious misdemeanor in Iowa. I suspect all states have similar laws on the books, just as a function of protecting ballot integrity. There have been instances of declared candidates being removed from the ballot, by a state agent.
I do agree that there is no federal offense in running while ineligible. Election contests are run by the states, and only in the case of Pres/VP are the ballots (those being ballots of electors, not of voters) counted by an arm of the federal government.