Fuller has some weird obsession with the real life Paul Stamets. I remember happening to look into an episode of Hannibal a relative was watching involving some killer named Stamets who planted mushrooms on his victims or something and making the connection with Trek's Stamets using a mushroom drive and then realizing both were produced by Fuller. It's just kind of odd that this fixation with Stamets and his mushroom work of all things would be so noticeably obvious across two tv shows.On paper Fuller would be a good choice.
However, he's also shown that when he's in charge, things can get rocky. his shows are often very esoteric, and short. And he really chafes at following someone else’s IP (Hannibal, American Gods).
There's no perfect equation. Nick Meyer, famously, knew nothing about Star Trek before helming Wrath of Khan. But these days, after 13 movies and 10 series, it would be foolishly to hire someone with zero knowledge of the franchise or genre.
It would be like naming Galen Erso's character in Rogue One as Bob Oppenheimer or something. Anyone familiar with the naming would be pulled out of the story so quickly because it's just so fourth wall breaking.