Something like the Time Travel Ban requires large centralized governing bodies to enforce it. But The Burn not only destroyed those, it also created a resource scarcity based situation that would encourage separate parties to use Time Travel to solve it.
You're missing one key thing. What Book said in the season premiere was that all known means of time travel were already destroyed at the end of the Temporal Wars, well before the Burn happened. It's not just a "ban" -- the means simply does not exist anymore. They couldn't go back in time even if they wanted to. There just aren't any time machines around anymore.
Although, of course, there are logic holes with that idea. Theoretically, any warp-capable ship is a time machine, with the slingshot effect. But I'm partial to the idea David McIntee (our own Lonemagpie) put forth in the novel Indistinguishable from Magic -- if slingshot time travel were easy, people would be changing history all the time, so it's probably a lot more difficult to do successfully than TOS and The Voyage Home suggested.
Plus, of course, there are naturally occurring time warps and anomalies and such, plus the Guardian of Forever. But I would guess that in the wake of the time war, the Temporal Accords signatories took steps to eliminate natural time warps or make them inaccessible. And maybe they put some kind of defenses in place to shut down time travel attempts. (Readers of my DTI: Watching the Clock will know what I'm talking about.)
Whatever the reason, the intent of the show's writers is that time travel is not merely outlawed, but unattainable. After the Temporal Wars -- and before the Burn, so the galaxy's governments were still in place -- time travel was deemed so dangerous that all existing methods were destroyed or blocked. So by the time the Burn happened later in the 31st century, it was already impossible to use time travel to undo it.