Rental car with No Insurance - worried about liability

I and my father was recently involved in an accident with a rental car last night. Thankfully, no one was hurt, and it was a single-vehicle incident. The accident happened due to icy road conditions, and the car lost traction and slid off the road. The damage to the car is significant — the wheel broke off the axle, and there’s a lot of body and mechanical damage, but the frame is still intact.

The rental agreement is only under my name, and the car was paid for using my father’s credit card. We didn’t take the additional insurance offered by the rental company because the credit card requires you to decline those for their coverage to apply. The problem is, I’m only now realizing that for the credit card insurance to work, the cardholder (my father) likely needs to be the primary renter or at least listed on the rental agreement, which isn’t the case here. I’m really concerned that the claim might be denied because of this, given I'm not listed under my father's card either.

I don’t have personal auto insurance, so I’m worried that everything will fall back on me. I understand that rental companies typically hold renters fully liable for damages if they decline the optional insurance. We haven't contacted the credit card insurance line yet since the rental company hasn't gone through the claims, so I’m not sure what documents will be required or how this will all play out. For now, the rental company has already given me a replacement car, so that part is sorted, but my main concern is what happens with the damaged car and the costs.

I’m not made of money by any means, but if it comes down to it, I’ll take responsibility and figure it out. That said, I don’t want to assume the worst just yet. If anyone has advice on how to navigate this situation, especially if you’ve dealt with something similar, I’d really appreciate it. From what I’ve heard, rental companies often tack on administrative fees, repair costs, and even loss-of-use fees, so I’m trying to prepare myself for whatever is coming. If anyone knows how claims work in this kind of scenario or has suggestions on how to minimize the financial hit, or any advice at all, I’d love to hear from you.