Also the 'Federation' guy that greets Burnham is definitely a hologram, he flickers slightly when in shot. Perhaps the true Federation have had to go into hiding to maintain themselves for an awakening at a future date, a sort of take on what happened to the Vaadwaar in Voyager.
Hmm... maybe a Foundation-like secret society that works behind the scenes to restore the Federation, complete with its own Hari Seldon hologram? I'd like that.
Of course what I'd like the most was if this Shadow Federation actually turned out to be the real villains of the season because they spent so long in their ivory tower that not only did the galaxy change too much in their absence (just like there would be no point trying to restore the Roman Empire as it was in 1776), but they themselves had forgotten what the Federation was meant to be long ago. Perhaps a nice little conflict and contrast between a half-remembered Federation-That-Was and a possible new Federation-To-Come, built for a new era but based on and inspired by the ideals of the old one, just like how the Roman Republic and Athenian Democracy inspired the French and American Revolutions.