It would be fun, but I don't see how it resolves this problem. Actually, it makes it even worse: if many forces, that aren't friends with each other and aren't controlled from the same center fight against the same enemy, then most likely, this enemy is some kind of an imperialist bastard who wants to take over the world. It would imply that the dwarves are agressors, that the dwarves are bad guys.
The previous answer I wanted to say that everyone is fighting against everyone. Sometimes enemies become allies. But still, each race protects its territory from any other races. But since we act out precisely for the dwarves, we see everything that happens from their side.
If the main role were spiders, we would have thought that they were a universal evil)
Okay, so we have grimdark setting where every side is an imperialist scum that tries to take over the world. Again, it doesn't help. The fact that everyone is an imperialist scum does NOT make dwarves less guilty. Even if there is war of all against all, they could just defend their homeland, instead of invading territory of other fractions (for example, the spiders have nothing to do with invasion of goblins. And place where the spiders live is clearly not homeland of dwarves).
Then I have a good question for you, what will help?)
I think you need two things. A noble goal to justify invasion in areas of other fractions and reason for unification of other fractions, without making dwarves imperialist.
How about something like this?
Dwarves are a peaceful fraction, although they are always ready to protect their homeland and their people. They lived in peace with their neighbours (amazons and wolves). Their only military problem was occasional raids of bandits. But then, the Cataclysm happened. By use of extremely dark magic their world was merged with another world, with the world of the Empire. The Empire exhausted resources and destroyed ecology in its world, so it used magic to merge its homeworld with another world (by the way, it was NOT the first time when it used said ritual, it explains why its races are so diverse), full of life, resources and potential slaves. The Empire used force to unite various races that hate each other (like orcs and spiders), using "divide and rule" strategy. Said races don't hate dwarves, they are just slaves of the Empire. As the only fraction from their world that has the biggest military potential, the dwarves were forced to defend themselves and the neighbours against the Empire. Against all odds (or maybe because forces of the Empire were overconfident) the dwarves were successufull at defending their homeland. But some of dwarves (including women and children) were caught by the Empire and turned into slaves. In order to free them dwarves gone beyound their borders. They won fight with slavers of the Empire. Then they wanted to make peace with the Empire after that (although trying to make peace with the Empire is just as stupid as trying to make peace with cancer. It's a parasite, it can't live without destruction.), but then their mages noticed something strange: out of sudden grass turned yellow! Many small bugs died. Whole anthills just dropped dead. They concluded that it can mean only one thing: something drinks life energy from their area like crazy. Animals and plants were sacrificing themselves in order to protect the dwarves (the dwarves are very close to nature, almost like elves in other settings) And this "something", according to their research, is located in the heart of the Empire. Instead of conquering them the Empire has decided to drain life energy from the dwarves and their homeland with some kind of dark magic ritual and then get hold on their resources (and possibly enslave survivors, if there will be any). The dwarves must act quickly and reach the heart of the Empire before they become too weak to protect themselves.
If you are worried about some kind of global goal. Then don't worry, it will.
However, the orcs and bandits here are negative characters. Orcs put on the village, and the bandits are called bandits for a reason.
The only race where you can empathize is the spiders)
But at the cave, the gnomes count as acting under hypnosis.
I have additional options on my mind.
1."Plot? What plot?". Just missions, without any attempts to explain why dwarves are fightings with such and such enemy.
2.Rebranding. Making dwarves openly imperialist and emracing it.
"But at the cave, the gnomes count as acting under hypnosis."
Wait, what? Who hypnotised them to attack spiders?
Dwarfs simply go to the mountain for gold, and the enemies do not allow them to pass and reach treasures.
This is not a good excuse. If the territory belongs to other fraction, then it has every right to forbid the dwarves from passing through their territory.