The thing is, as far as was shown and hinted at, the Federation wasn't fighting an offensive war. Except for what the Discovery did before disappearing to the MU, it was purely defensive. And defending against 24 separate factions that attack at random, at random targets, is a struggle to say the least, especially with mounting losses. And even though the Klingons had no single leader, they did have a single enemy that they were attacking, so in that they were joined, even if not organized.
I dislike the 9 month time frame as well, but there is a modicum of logic in the premise, even if it is stretched to the limit.
Analyzing Starfleets strategic decisions in Discovery is complicated by the writers total failure to understand basic fighting tactics.
If I sat at the Federation strategy table, my first objective would be to decapitate every clan leader in the ranks. Knocking off 24 House heads probably isn’t realistic,but taking down even six means six armies out of action temporarily, while the other 18 house leaders would either have to hole up somewhere or risk the same fate. We know Starfleet has several classified projects in the works like Discovery, so a hit list like this wouldn’t be difficult.
In fact given the spore drives capabilities, USS Discovery could pull off a decapitation strike against the Klingons perfectly. Jump in ,blast the flagship of each clan,jump out.
Then there’s logistics and energy production targets on the Klingon side. Praxis would be a juicy target in a UFP/Klingon war.
All of that is to say an intelligently run fight from the Federation means offensive missions in Klingon space . If they just park ships around space stations and wait to be attacked it’s a given they’ll lose. Unfortunately I doubt the writers understand that , so here we are.