People may chose an ineligible person due to lack of information, perhaps intentionally withheld; their choice does not make an ineligible person eligible.
This is not a democracy, this is a republic. Rule of law, not rule of majority.
Put another way, popularity does not launder ineligibility.
Which is why the 20th Amendment says that a President elect - somebody who has received the electoral votes necessary to become President - cannot ACT as President if they have failed to qualify. The Constitution itself has the power to keep an unconstitutional President from taking any Presidential action, even if the people have elected that unconstitutional person. Same thing with a Vice-President elect.