He has also drafted celebrities who, he says, don’t approach the issue with “political baggage.” For instance, he recruited former basketball stars Lisa Leslie, a three-time MVP award winner in the WNBA, and Jalen Rose, now an ESPN analyst, to record public-service announcements backing school choice. Other messengers include black civil-rights and church leaders. Last year Mr. Chavous helped organize a rally in Memphis with a keynote speech by the city’s most prominent minister, Rev. Dwight Montgomery.

Unions tell parents “that this is a Republican conspiracy,” Mr. Chavous says, or “that you all are trying to destroy neighborhood schools.” When low-income, working families are “told bad things, they initially fight. But when they understand what this is all about, I’ve heard so many parents tell me, ‘Well, you didn’t tell us . . . I can get a scholarship to the Catholic school, this private school that I’ve been wanting to send my kids to for years, if this passes.’ ”

School-choice opponents, he says, use a “playbook,” which entails, among other things, outright lying. A typical accusation is that choice advocates are bought by the billionaire Koch brothers. Opponents also claim that vouchers don’t work, but then struggle to explain the high demand among parents who they then claim don’t know what’s best for their kids.

There have been few controlled studies, Mr. Chavous notes, but data generally show that the children who receive scholarships benefit educationally.