ADEL, Iowa—Democratic leaders in Iowa, the starting line for the party’s wide-open 2020 presidential contest, are hungry for a young standard-bearer who will usher in generational change, which is erecting a potential roadblock for the three best-known prospective contenders for the nomination.

There is little consensus about which issues should define the next nominee. A Wall Street Journal survey of Democratic chairmen and chairwomen in the state’s 99 counties also found support for a contender who can appeal to the ethnically...