Because if you go directly to their insurance it will be just you against their army of lawyers. If you use your own insurance it will be your insurance's army of lawyers versus the shop's insurance's army of lawyers.
The technical insurance term for this is subrogation. You report to your insurance company, they pay you for the damage (minus your deductible), and then they subrogate against those actually responsible. When they recover for the damage, they pay you back your deductible.
Now, this requires that you have physical damage coverage on your Auto policy, and not everyone does, but it can be very effective. Your insurance company is incentivized to want to keep you as a customer. The dealership's insurance company DGAF.
The other option, now that you know that's a possibility, is to raise it to your dealership that you're going to report the damage to your insurance company, and in doing that you ask for the carrier and policy number for their "Garagekeepers Liability Policy". The dealership may be more incentivized to keep this out of insurance if they think claims will raise their premiums.
I suppose you can always fall back to your own insurance later if you can't get them to play ball.
If this is your plan, check your policy. There may be a requirement around prompt claim reporting, but that's highly state-specific on personal auto policies.