It does. But what I think Billy should be is, well, innocent.
Now I dont know anything about the early Captain Marvel stuff, but I suspect it was nothing like how Johns rewrote it all as Billy being angsty and lonely fosterhomer. He may have come from a place like that, but back then it was a time when he could turn a corner an bump into a friendly tiger walking upright and dressed in tweed.
This is kinda why I compare him to Teen Titans, Billy almost should be like the cartoons in this regard. You have have serious moments (not gritty darkness), but remember to have an equal amount of fun.
To be honest, I dont think Deathstroke outgrew the Titans at all, I think alot of DC writers simply realized that he was a great character you could do things with. Especially since Deathstroke broke off around that 'magical time' where everyone was getting into the extreme phase.