Yeah, like I said that would appear to confirm that the breaking point was much more recent that one would think, but it still doesn't quite add up. If the Republic knew about At Attin right up until the end of the Clone Wars; why didn't Palpatine take control of it, just like he did everything else? Also a single generation doesn't seem nearly enough time to allow for both Tak Rennod and At Attin to have entered into the realms of myth and folklore.
What makes the most sense to me is that they've been radio silent for much longer, and the Supervisor was just continuing to monitor an official government comms channel for bulletins as per his programming (possibly awaiting a specific all-clear code that was never transmitted), and the last message to go out over that old frequency before the Empire shut it off and switched it over to something new was Palpatine's declaration.
As for why specifying that connection at all; clearly it's just a convenient way of blowing Jod's cover. The show doesn't really have to elaborate on the details since all of the witherto's and whyfor's aren't really terribly relevant to the story. Indeed I rather appreciate how restrained the exposition has been on this show as a whole!