There are several overlapping issues that have been touched upon in this thread. It's like trying to unravel a ball of yarn. Careful! You'll wind up like a kitty getting rolled into it.
Tracing back how the Borg were originally defined to us:
Guinan told Picard that the Borg were a conglomeration "of biological and artificial life that has been evolving for thousands of centuries". How Guinan came to know this is unclear, but if we are to accept this basic premise as the foundation, something does become clear: the Borg have existed the same Galaxy as Earth and the Federation for at least a quarter-million years and possibly longer. Guinan also told Picard that the Borg "swarmed through our system" and destroyed the civilization on her homeworld.
After Q's encounter at J-25 had been broken off, Guinan also told Picard that Q introduced the Borg to the Federation at a time long before the Federation was ready, and "for now, for right now, you're just raw material to them", but she also said that the Federation could establish a relationship with the Borg once they were prepared.
Consider a comparison between Gomtuu (TNG's "Tin Man") and the Borg. Both a Borg collective and the Gomtuu space vessel appear to be some form of "living" space vessels. Both are capable of hostile acts that a Federation starship captain cannot reliably anticipate, diffuse or defend against. (Witness what happened to the Romulans.)
Of course, the Borg are "bad guys". They are mean as mean can be. But also consider this: another important difference is that Gomtuu, to our knowledge, had no history of aggressive violence. Sure, it could be provoked into destroying a Romulan Warbird or crippling the Enterprise-D. And Gomtuu's moves were obviously not logical to Picard. The Federation could easily regard Gomtuu as a threat. Would Starfleet put an all-points bulletin out for starships to avoid, or even to take defensive action regarding Gomtuu? Not likely. Gomtuu would be seen as a potential threat, but also as a first-contact mission. Starfleet would want to assume a risk.
Same with Species "8472". These aliens might seem like a super-threat even worse than the Borg, since a fleet of their bio-ships is capable of Death Starring a whole planet. And, like the Borg, 8472's actions are clearly trending to the aggressive end of the scale. Does the Federation regard them as a threat? Sure. But they are not a direct one.
But the Borg have attacked Federation starships and allied colonies and installations. They tried to invade Earth twice, and have assimilated Federation citizens (presumably by the thousands, perhaps more). Surely, the Borg deserve an even stronger response, do they not?
Indeed they do, but consider this as well: the Borg have been in existence for at least a quarter, maybe a half-million years or more. If they had been so aggressive from their very beginning, openly marauding the Galaxy for all those "thousands of centuries", then it's safe to assume that there would not be anything left for the Federation to explore.
Clearly there is a Galaxy to explore, and therein lies an undefined issue with the Borg. Have they always been so violent? If they had, then the Galaxy would already be overwhelmed by them. Clearly, there are unknown aspects to Borg behavior. Do they go through periods of violent aggression for some unknown reason, only to stop at times and become peaceful? Or were the Borg once peaceful/docile, and something happened to change them? Guinan seemed to think that the Federation and the Borg would eventually have a peaceful relationship. Could it be that the Borg have encountered other distant powers who have/had a relationship with the collective, and causing the collective to set aside their violent behavior? Or did the Borg only acquire sufficient technology to travel through deep space only recently, with nearly all of their long history being confined to a single planet or sector of isolated space?
Looking at the Borg from this perspective probably gave the Federation leadership reason to pursue a more cautious path. The Borg are not a conventional military threat. Therefore, they defy conventional military and diplomatic thinking. As Q pointed out "you can't destroy them." And even if you could infect thousands of their ships with a computer virus, you can never be certain that the virus reached all of them (some may be out of range) or that the Borg will find some way to adapt. (Surely their collective cyber-consciousness network has built-in security countermeasures.)
So maybe the notion of "let's declare war on the Borg" would be seen in the same light as "let's declare war on Species 8472", or "let's declare war on Gomtuu". Such a move would be seen as Quixotic and even counter-productive.
The bottom line is this: it's never wise to declare war on an adversary unless you have a clear strategy that leads to victory, and you're willing to deal with the aftermath. Just ask Hitler and Tojo how that works out.
There's also the old saying "Your enemies of today will be your friends of tomorrow", in other words, before you go getting into a fight with that guy at the corner bar and you decide you're going to go all-out and break his arm to teach him a lesson, you might want to consider the consequences of all-out warfare. Maybe it's best not to break the guy's arm. If you avoid him, he won't be likely to try to smack a pool cue over your head.