WASHINGTON—If everything goes according to his plan, Pete Buttigieg will celebrate his 39th birthday on Jan. 19, 2021, and the next day, raise his right hand to be sworn in as the 46th president of the United States.

The South Bend, Ind., mayor’s youth and outsider appeal is proving an asset with some primary voters who appreciate the contrast he draws with 73-year-old Donald Trump and who want a candidate they believe is up to the rigors of two terms in office.

But...