It's hard to gauge at this point if that's actually a plotholel or not. While that is stated, it's also stated that "only gods can time travel" using them as well. It could be that the usage of the time rings isn't tied specifically to the position of Kaioshin, only that Kaioshin are the ONLY ones that are supposed to use them and no one else as in it's some type of universal law of some sorts (such as time travel itself being banned outside of the usage of Time Rings).
In the anime, the only thing that makes Black and Future Zamasu into Kaioshin is them taking Gowasu's potara earrings, though that's never depicted in the manga in any capacity. Though it is stated in the manga when Gowasu gives Zamasu his potara to travel into the future on Planet Barbari that he's "temporarily promoting him to Kaioshin" which means that there is probably more to becoming a Kaioshin than just taking the earrings IN THE MANGA. Dunno what was stated in the anime though.
Dragonball is bit ambiguous here on this matter. It could be interpreted many different ways and it's likely Toriyama himself was rather vague on the rules as well.