Earth is not likely hostile to the Federation as it first appears in the episode. Not only did they allow Admiral Tal to continue to operate on Earth, they allowed the joined entity that took his symbiont to become part of Earth's defense force. Moreover, I would guess that Adira was not present on Discovery at random. She was placed into the inspection team specifically because the symbiont had knowledge of older Starfleet technology. It may even be that Earth is hoping that some effort to revive the Federation succeeds.
I posited that when the Federation HQ departed Earth, they may have concocted a plan so that Earth 'officially' withdrew from the Federation (to avoid it being a target) and of course to preserve it for eventual return once the cause of the Burn was discovered and dealt with... but otherwise that Earth may be in on this plan intentionally.