The very first S31 episode had Starfleet Command actively stonewalling Sisko’s inquiries, and Kira openly calling it a cover-up, so no, I don’t think its status as part of Starfleet was ever under serious question.
It was still unknown whether it was Stafleet, or starfleet individuals supporting it illegally. From Discovery, for example, everyone seems to know about S31, and official starfleet processes go through S31 as a rule, making it part of the starfleet structure and not just an illegal (even if supported) group
In what way was it unknown? Sloan had the active support of Starfleet command, access to at least one starship, and full authority to do whatever he wanted. None of this is ambiguous, and is in fact directly stated.
The way I saw it, it felt like they were saying that starfleet command is corrupted and using S31 as a way to get things done no matter how illegal they can be. Which is why when getting caught they would deny down to the existence of it all.