What makes you think that the Andorians and Klingons -- species that have long histories of partnership with Humans, species that have fought and died beside Humans, one of which has actually yielding its independence to forge a political union with Humans -- would just betray Humanity because a few Humans and one Caeliar happen to have been the origin of the Borg? Why would they do that, especially knowing that the Borg have consistently targeted Humanity before any other species for enslavement or extermination?
Because these are species that love to fight. Especially in the Klingons' case - they will certainly look for *any* excuse, no matter how absurd it may seem to us, to fight. Their entire culture is based on it! They are warriors - that's all they do.
THAT is a gross oversimplification of the Klingon race. Yes, most of the Klingons we see do love to engage in combat - Klingons are a durable race. More importantly, though, anyone outside the empire usually sees just the warriors, not the farmers, actors, thinkers, etc. of the Klingon race.
The Klingons have in the past been beaten down and they stop with the warfare long enough to pick up the pieces and rebuild. They did this after Praxis, and the Chancellor then was much like Martok in the matter of his respect for humanity.
At this point, I think even the most violent and outspoken species are more concerned with licking their wounds than entering into another costly war. Every one has taken heavy blows here.
And, on top of that, how CAN they punish anyone? Okay, so the Tholians can hold a grudge for multiple centuries. But what happened with the Solumbia crewmen and Sedin was both three centuries and six plus millennia ago. Yes, the 'blame' can be traced back to the Columbia mutineers, but if you're going to backtrack that far back, why not just blame the Romulans for launching the attack on the Columbia in the first place?