Starfleet originated on Earth before the Federation was born. Why would they ever leave, even when the Federation disappears?
Because they don't agree with where the Federation is going. They're afraid of the Burn happening again and become more protective. Because some time traveler told them bad things would happen if they stayed. It's Trek. There's always a reason.
Or perhaps the other way around - maybe the other worlds got tired of the "Homo Sapiens Only Club" mentality from humans and decided to secede.
Without a real fleet to defend it, Earth might have seemed an indefensible position. Without Starfleet, Earth might have taken a natural course of action and reverted to defend itself. The food shortages alone without using enough replicators would have been staggering. Burnham might have only been calculating deaths in starship losses. World losses might have been higher. It would be interesting to find out just how big the federation was when everything went boom. Were Klingons members of the UFP? Presumably the Xindi were. We don't see Borg hauling ass, so either they were also affected or were already extinct, or they just found nothing left worth assimilating.
^^^ Somebody on the ST: D VFX team put an Easter Egg in that shot as one of the supposed Federation ships looks like the Space Jellyfish Creature 'ship' from TNG's pilot episode "Encounter at Farpoint" before it transformed into it's more Jellyfish like state (It's the 'saucer-like' ship on the second row, right side) :
It looks far more like a mothership from the ‘V’ miniseries. https://vignette.wikia.nocookie.net...attle77.jpg/revision/latest?cb=20160325234249
Curiously, it doesn't appear that any of the elephant ships of UEDF from this week's episode would be part of the kaboom fleet. The ship right above the "mothership" may be related, but it's not quite the same design flipped or anything. Timo Saloniemi
I'm betting not. They seemed to have thought about that. The dilithium shortages started around the 2600's, from what I could gather from the dialogue, so there were several centuries to deal with shortages. It mentions they tried many alternatives for ships but it had not worked out. My guess is since dilithium was needed for warp travel, non warp travel powerplants had probably already either been converted to some other power source. We saw the return of solar panels along the Golden Gate Bridge, so they may have been using renewables before the Burn. And there's always fusion as well, which does not require exotic elements. That could also mean why deaths were "only" in the millions. If they had stopped relying on dlithium to replicate everything, that would have prevented trillions from dying due to famine.
Ah, that's because Enterprise was written before Discovery. They didn't actually have a capacity for time travel when ENT was written so they couldn't know back then that fans would obsess about canon so much. Otherwise they surely would have obeyed fandom's wishes a lot more.
Starfleet stopped being an Earth organisation from about 2161. It would’ve had much stronger affiliation to the Federation, by the 31st century. None, presumably. Time travel was outlawed, at that point.