It's weird too because I have trouble imagining the insurance company is happy footing a repair bill (minus deductible) when the other company should be paying?

Like when OP goes to their insurance and says "hey bud my car needs repaired" aren't they gonna ask how it happened? And then contact the other insurance?

Hell I would worry about the insurance company denying to cover anything and saying policy was somehow violated by making an agreement that might save the other insurance company from paying damage they should