What I really want/need is some mention of the Dominion. With the Federation in tatters, surely that would have been a huge opportunity for the Dominion to muscle back in?
Why assume the Dominion is in any better shape than the Federation? The Burn was galaxywide. If it had been targeted at only one part of the galaxy, then dilithium could've been replenished from elsewhere, or else the remaining unaffected powers would've long since moved in and either conquered the affected ones or helped them recover. So we can safely assume that every widespread, warp-dependent interstellar power in the galaxy was crippled or shattered altogether.
If anything, the Dominion would have been far less likely to survive the Burn, because it was an oppressive empire dependent on the Jem'Hadar's subjugation of the client worlds. With warp travel hobbled, the Founders wouldn't have been able to maintain their iron fist or put down insurrections. If the Dominion even still existed at the time of the Burn (700 years is a long time), it's profoundly unlikely that it would've survived long thereafter.