To be specific, what you said was the the Akatsuki used the Hidden Mist as their base of operation even though there isn't a shred of evidence to this claim. Furthermore it doesn't make much sense as we know that the Akatsuki as a group moved around and had no one given location as their home base. Hence why they referred to their meeting locations as "Hideouts" as in the plural form. They had several bases, not one and each member seemed to either prefer to travel or have a home location of their own.
Obito for example remained in the cave where he met Madara, where as Itachi retreated to the Uchiha Hideout. You didn't say they believed it was their based of organization because of rumors, you said that the Akatsuki grew their organization in the Hidden Mist which we also know to be false as they were a group that took on contracts from different villages such as the Hidden Stone.
As stated above, there isn't a shred of evidence that proves the Hidden Mist and Akatsuki in cahoots and that they used the Mist as their base. We do know that they had several bases and that Obito, Nagato, Itachi and other members had separate bases of their own.
Now you're not only contradicting your own statement, but you are bluntly put inaccurate. We see the formation of the Akatsuki controlled by Obito and its formed in the Hidden Rain after Nagato agrees to join him. Hence the location of their formation is in the Hidden Rain, not Mist.
The former akatsuki that nobody knew about started in the hidden rain. Nobody was talking about that version of akatsuki to begin with and that’s why I said that you are talking about two completely different organizations. You earlier said that the Mist being the birthplace of the Akatsuki was nothing but a rumor, hence the contradiction as you are now claiming it as indisputable fact.
Summary
- Despite your claims there is no evidence of Obito's Akatsuki originating in the Hidden Mist. It was just rumors.
- We don't know when Obito took control over the Hidden Mist, hence we don't know if it was before or after he formed the Akatsuki with Nagato and Konan. Though we know that at the point of him controlling Yagura he no longer needed GuruGuru like he did when he formed the Akatsuki with Nagato and Konan.
- The Akatsuki had different bases of operations referred to as "Hideouts" and often traveled to keep their locations, members and plans a secret.