Words mean different things to different people is my response to that

. Me saying it's Captain's pet project isn't really saying much more than what you did:That she's probably included on meetings. It could be others in the community really do think she has all the power, but that's not what I think.
Like, do I think she'd ever have the final say on things? No, but on the other hand, I do think that if she said something, or suggested something, it would be part of their conversation about the girls and not just ignored. I'd even truly guess that she had quiet a bit of say in which girls got brought over to Tsubaki (though not the final decision), they've never made it seem like Maimi had a say in anything that she's announced, and I don't think Saki would post things about why she thought certain girls would do well in the group on her blog if she had had zero say in the decision. Is she a producer of the group? No. Is she even close to being a manager, do I think? Of course not. I just think she's an adviser and not just a mouthpiece. Saki does have over 10 years of experience being an idol in H!P and they wouldn't just pay her to stick around and be a "face" for announcements. She's not around nearly so much for that to be it, in my opinion.
There is a point though that it might just be a KSS thing. It could be the moment they're debuted Saki won't deal with Tsubaki pretty much at all, like with Kobushi. It's that "Factory" connection that tends to make people think differently though.