If the name “Majority Forward” sounds familiar, it shouldn’t—transparency is not the group’s style. But perhaps no organization had a bigger impact on securing, for the first time in years, the Democratic Party’s thin control of the U.S. Senate in 2021.

Majority Forward, the 501(c)(4) arm of the Democrats’ partisan Senate Majority PAC, is a “dark money” behemoth. You wouldn’t know it from the group’s nondescript website, but Majority Forward spent $515,000 backing Joe Biden in 2020 and pumped $1.7 million into the January special elections for U.S. Senate in Georgia to boost Democratic candidates Jon Ossoff and Raphael Warnock. Their victories decided control of the Senate.

But Majority Forward’s main value to the left is serving as a pass-through to other political...