I'm not sure you understand why DC came to rent Captain Marvel from Fawcett in the first place.
They did it to spite Marvel Comics. Marvel used to be called Atlas (and before that Timely); it had been Marvel since 1961, and published its own Captain Marvel for the first time in 1967.
DC (as National Comics) had forced Fawcett Comics to stop publishing Captain Marvel in 1953 (arguing that Captain Marvel was just a Superman expy, even though they're fairly dissimilar and Captain Marvel was at that time far more popular). Fawcett remained in other areas of publishing, but didn't bother to do comic books again.
In 1973, DC rented the Fawcett comics catalog; this included reprint and other rights (such as tv shows). They immediately began exploiting those rights - and sued Marvel Comics over infringement of the Captain Marvel trademark, only to find that Marvel Comics owned it and had done since 1967. Because Fawcett hadn't bothered renewing their trademarks (and had been out of print, so had no reason to do so), DC had no legal basis for their claim. In effect, they'd paid a lot of money to spite a competitor, only to find they'd fucked themselves 20 years earlier.
It was only in 1977 that DC bought the Captain Marvel (and the rest of Fawcett's characters) rights from Fawcett; by then they'd agreed never to try and use the Captain Marvel name in any promotional material again. Which is the reason that they don't, and all the shows are named Shazam, and the movie is named Shazam, and even the character has been officially named Shazam since 2010.
The reason he barely gets any material is that he's unpopular.