Something to bear in mind is that even if Starfleet Academy wasn't functioning anymore, the remaining Federation Member planets might have had space force academies of their own that could have provided comparable training. The
Virginia Military Institute is an organ of the Commonwealth of Virginia, not the U.S. federal government, but its graduates may receive officers' commissions from any U.S. Armed Forces branch except the Coast Guard. And we know from S1 that graduates of the Vulcan Science Academy could receive Federation Starfleet officer commissions as far back as the 2250s. So there's a strong chance that Member governments' own planetary space force academies were feeding what remained of the Federation Starfleet officer corps after the Burn.