Washington, D.C.,’s once-beloved Metro subway system isn’t great anymore. It isn’t even good. The agency’s slogan, Back2Good, acknowledges as much.

To win back riders and put high-profile safety problems behind it, the general manager of the Washington Metropolitan Area Transit Authority, as Metro is formally known, says it needs an annual $500 million infusion of new capital funding from Maryland, Virginia and the district.

Elected...