I see Johnson as Adam much more than Captain Marvel/SHAZAM. That's me. Could he play the hero part? Sure. Beyond just the wonder and naivete of Cap when it's Billy's personality at the helm the character also lends itself to humor which Johnson has the chops for. If they go with the whole Cap is a separate entity take I could see Rock playing up the whole Dudley Do Right aspect as well. Granted I've never understood that version of the character. Billy transformed makes a certain amount of sense. There's too many unanswered questions in the other take. What happens to Billy, exactly? This separate person that is Captain Marvel, does he have a history, or does he just come into being when the lightning comes down? Does he go somewhere else when Billy comes back? If so, where? There's a strangeness to that take I don't think even young kids today wouldn't find odd. Billy in control of Marvel's body is the way to go.
As for casting Billy, I'm not sure as to whom could do it off the top of my head. I think it should be an honest to Zod kid. Under 14 but older than 9. And they have to handle their lines well, so perhaps a child actor with some film or TV credits to their name. Complexion or race aren't a biggie for me, but maybe a kid that has physical features similar to the Rock if he is SHAZAM wouldn't hurt.
I still think the best damn version of Marvel/Shazam was that Superman team up short. They even had a respectable Tawney in that. The vibe of that short is overall a good template.