So you're saying that producers don't sometimes have a fairly large part in the creative aspects of a game and overseeing the whole thing? That's horseshit. I doubt most producers are real hands-on and are more out to gather programmers and other talents and that kind of work, and to be the guy who mentions to the designers and lead programmers, et al. what the big-time executives are looking for—but if you don't think Miyamoto is much more involved than that with some of the bigger Zelda and Mario games, you're kidding yourself.
Just like with music and movie producers, there's many different roles a video game producer has. It's not a standard term that means simply one thing. It's actually a bit of a vague term.
Now how much involvement Koji Igarashi has with the creative and design aspects of Castlevania, I have no clue.