The Insulation, for starters, SHOULDN'T be 'blocking' the vent. Go ahead and pull it away from the vent (via the attic.) If the insulation is covering the vent, your attic isn't being vented. With that, I'd replace the damaged vents and tack some 1/2" wire mesh on the attic-side of the vent. The birds shouldn't be able to pull that mesh or 'break' that wire too easily. If it takes too long, they'll move on. Again, move the insulation away from the vent, replace the vent, and tack some 1/2" wire mesh up there (in the attic) over the vent hole.
Yup, remove the 'source' of their nesting material and they should go away. My best to ya and hope this helps.