John Brackney, a former county commissioner from the Denver suburbs and registered independent, will be able to do something for the first time on Tuesday: cast a vote in a Democratic presidential primary.

A pair of 2016 ballot propositions changed Colorado’s presidential caucus system to a primary for the first time in 20 years, opening the nominating process for both major political parties to those who aren’t affiliated with either—more than one-third of the approximately 3.5 million active registered voters there, according...