On September 9, Colorado Secretary of State Jena Griswold appeared on MSNBC and said that the 14th amendment, section 3, likely gives her the authority to bar former President Donald Trump from the Colorado Republican presidential primary ballot. See this story.
The 14th amendment, section 3, does not give any government official the authority to block voters from voting for whom they wish. It provides that certain persons may not “hold any office, civil or military, under the United States, or under any state”, but it does not say that the government has the power to block voters from voting for such a person.
When the 14th amendment was written, voters had complete ability to vote for anyone they wished, because voters were permitted to prepare their own ballots.
The 14th amendment, section 3, also says Congress may by a vote of two-thirds of each House remove the restriction on an individual. It does not set time limits for Congress to do this. It is feasible that if the candidate were elected, congress could hold such a vote after the election but before inauguration day. But if state officials could block individuals from competing in an election, that would short-circuit that process.
Voters in the 19th century had more freedom than voters of the 21st century. This shrinkage in the power of voters has occurred almost without anyone noticing. There are very few scholarly articles that discuss how voters have lost the fundamental right of free choice. When the U.S. Supreme Court ruled in 1992 in Burdick v Takushi that there is no unlimited right for a voter to vote for anyone, there was very little criticism of the decision.