Well, of course. But Trek has been retconning things ever since James R. became James T. and UESPA became the Federation Starfleet. We only pretend it's a coherent whole.
Don't get me wrong, I wish the Klingon War in DSC had been of a more moderate size, since its Dominion War-level magnitude is hard to reconcile. But it's what we have to deal with now, and we just have to make the best of it, like we've been doing with continuity problems in Trek for nearly sixty years. There are many things in Trek that I wish had happened differently, but you can't relitigate the past, only deal with the present.
Maybe not a war of the magnitude we saw, but it's hard to believe there wasn't some warfare in the pre-Organian part of that "almost seventy years of unremitting hostility" that Spock mentioned in TUC. After all, Kirk considered himself a soldier, and he and others in Starfleet clearly had significant combat experience. Why not against the Klingons, given their proclivities?
I don't see any reason to think that. After all, it's no different from WWII being "round 2" of WWI. Wars often have sequels if the diplomats and politicians fail to resolve the underlying issues adequately. I mean, we know the war in "Errand" began with the breakdown of negotiations. That could easily be taken to mean that the years-long negotiations to establish a lasting peace after the previous war fell apart, and the war resumed.
Maybe this isn't what you're thinking, but if the question is "Why didn't the Organians stop the first war?," the answer is right there in "Errand": The Organians aren't peace activists, they just can't stand the presence of corporeal beings and wanted the noisy kids to get off their lawn. The only reason they intervened was because the war came to Organia. So as long as the previous war didn't affect the Organian system, there's no reason they would've done anything about it.