There is a real life example which is better than a hypothetical.
In 2004, Róger Calero, who was born in Nicaragua and is a naturalized U.S. citizen was the Socialist Worker Party candidate for President of the United States and received 3,689 votes.
Because he is not a natural born citizen of the United States, Calero is ineligible to become U.S. president under the United States Constitution, meaning that even had he won the election, he would not have been permitted to serve.
James Harris, the Socialist Workers’ Party presidential candidate from 2000, stood in on the ticket in nine states where Calero could not be listed on the ballot and Harris received 7,102 votes.
Róger Calero again ran for President in the 2008 presidential election and again, James Harris stood in for Calero in several states where not being a natural born citizen prevented him from even being listed on the ballot. In the 2008 presidential election, Calero was allowed on the ballot in five states, where he received 7,209 votes and James Harris received 2,424 votes in five states where he was on the ballot in place of Calero.
Since the courts have ruled that Barack Obama is a natural born citizen, the point is moot as to whether there is a difference between a “citizen” and a “natural born citizen.”
The courts have consistently ruled since the 19th century that anyone who qualifies as a “Citizen of the United States At Birth” is also a “natural born citizen.”
That is true for 14A citizens at birth. For everyone else, the issue remains unresolved per the State Department.
7 FAM 1131.6-2 Eligibility for Presidency(TL:CON-68; 04-01-1998)
d. (snip) In any event, the fact that someone is a natural born citizen pursuant to a statute does not necessarily imply that he or she is such a citizen for Constitutional purposes.