Yes, I can help. I'm in auto claims. Yes, you absolutely have a right to fix the car where ever you choose. I'm suspecting the plow guy is trying to handle it out of pocket. Insist they report it to their auto carrier. Yes, they should pay for a rental. Hidden damage should be found by the shop of your choice, if they are worth their salt, they'll find it.
Should she call her carrier as well? Well, that depends. If she has collision coverage, she can. They would then pay for damage (less her deductible) and pursue the plow company insurance for reimbursement. However, if she has liability insurance only, then no, she really doesn't need to report it. Given its age, I'd be concerned with it being a total loss. In that case, you can buy the salvage back, but be sure to look into whatever process you'll need to follow to get a salvage title for the car...it varies a lot state by state.
Also, you could ask Aamco to file a claim against their insurance called "Garagekeepers". The car would be covered there too.
Yes, the policy language says all claims should be reported. But honestly, if your car is hit while parked and unoccupied, and you have NO collision coverage then there really is no reason to report that. And as far as paying them to handle any incidents that are not your fault...well, sort of. For example, yes, if someone makes a claim against your policy and you aren't at fault, they will protect you. Or if you are at fault, of course their job is to indemnify you. However, if you are sitting at a stop light and are rear ended, but have no collision coverage on your car, they will not go after the other parties insurance for you. They could do you a favor and file the claim with the other carrier to help you get started, but from there, they won't do anything. In this case, if OP doesn't have collision coverage, which I suspect they may not, then her insurance will not do anything to pursue the plow guy...they can't, they have no legal standing to do so.