I mean, take what you will about Tecmo as a whole, but the Ninja Gaiden and DOA IPs have seen better days. It's not like we can't even feel Itagaki's absence from there or anything like that. That's the other side of a coin; a director may just be one team member, but that doesn't mean he's someone that's easily replaced, so one leaving isn't something to take lightly.
Also, that statement about Miyamoto is rather bold when you consider that there actually were notable names who jumped ship from Nintendo. (Intsys's Shouzou Kaga, or the guys that made Armature Studios, for some examples)
The idea that Miyamoto in 1991 would somehow do worse than any of those would be rather farfetched, I'd imagine.
Of course, the art of founding a whole new studio is a good bit of a different beast than just directing one game under another company's employ, so all of this is more complicated than that. But regardless, there's Igarashi and Kojima-likes to contrast with the Inafune and Itagaki-likes, so clearly there's something more going on that just "Director is important" vs. "Director is not important".