When Leah Daughtry, a former Democratic Party official, addressed a closed-door gathering of about 100 wealthy liberal donors in San Francisco last month, all it took was a review of the 2020 primary rules to throw a scare in them. Democrats are likely to go into their convention next summer without having settled on a presidential nominee, said Ms. Daughtry, who ran her party’s conventions in 2008 and 2016, the last two times the nomination was contested. And Senator Bernie Sanders is well positioned to be one of the last candidates standing, she noted. “I think I freaked them out,”...