The vast majority of people now, like the vast majority of people at the time of our founding, have never heard of Vattel or what a relatively few people sincerely believe would have been Vattel's thoughts on the "natural born citizen" standard if he had ever thought about the matter.
A commitment to popular elections rests in part upon a belief that sound and sensible decisions will emerge from a democratic electoral process, i.e., that the consensus of 130 million voters is superior to the consensus of 9 elitist judges or 1 elitist politician. A few voters may attempt to channel an 18th century Swiss philosopher and a few may even turn to a ouija board. Fortunately, most will do neither.
A democratic republic may sound like a crazy idea, but it is the American idea. And, our Constitution is worth defending.
They are certainly free to vote for whomever they like.