The pigeon is a nonentity in the accident. It's as much at fault, or not, as a tree. And it doesn't need medical or cremation bills paid. Cleanup is handled about the same way as debris from a car.
In a situation where the dog or owner isn't negligent but the driver is, it's useful to talk about whether the dog was on the leash, in the crosswalk, has right of way, is walking with the walk signal on, and is visible. You might prefer to leave the dog out of it and talk about the owner, but I could see myself wanting to get straight to the point and talk about the dog. I imagine lawyers also might be inclined to cut to the chase and talk about whether the dog was following the rules of the road.
All those things, as you say, can bear on the driver's responsibility (or the owner's responsibility, if, for example, the owner allowed the dog to go out into the street when the walk signal wasn't on). But that's part of the factual description of the case. It does not assign fault to the dog; if the dog did something like go out into the street when the walk signal wasn't on, that makes it the owner's fault, not the dog's.
The only argument I have remaining is that while it's an incorrect statement, I don't think saying the dog is at fault in an accident is insulting, because being at fault in an accident is often only an honest mistake (albeit sometimes with tragic consequences).
To drive this point home, a pigeon dive bombing your windshield in the middle of an intersection could be correctly caused the cause of an accident. The pigeon is a free bird. It has no custodian.
Pets, on the other hand, are always in someone's custody. Custody relationships are complex, but ultimately come down to "you're responsible for this." Anything a dog could do to harm itself or others could have been anticipated, should have been anticipated and is ultimately the custodian's fault.