Driving across the flatlands of Illinois with Barack Obama during the Senate race of 2004, Kevin Thompson sometimes found himself tutoring the candidate on gay rights. Mr. Thompson, then a traveling aide, recalls long conversations about topics like the 1969 Stonewall Rebellion that sparked the gay rights movement, gay adoption — Mr. Obama once volunteered that Mr. Thompson and his partner would make “great parents,” Mr. Thompson recalled — and same-sex marriage, which Mr. Obama has in the past opposed. Mr. Thompson, an Obama supporter, is skeptical about that. “To this day,” he said, “I don’t think Barack Obama has...