AB Car Rental's
website shows two types of insurance. The "standard CDW" which most of us usually take, and what I think AMEX covers (other credit cards cover CARS, but not TRUCK, including Pickup Trucks). This coverage is only for damage to the rental car due to the fault of the renter (or damaged caused to a 3rd party) with a $350 USD deductible. For only $19 a day, you can get "Full Coverage" insurance, which reduces the deductible to $100, and also included "other" damages to the car by "third party, flood, fire, act of God, etc." This is a newer (to me anyway) coverage to try to cover damage by thief, along with other "natural" causes. I'm not sure if this would cover a tire or not, but it would still be subject to $100 deductible.
An interesting sideline to this is that rental car companies are required (at least in the USA) to carry liability insurance on their cars. So, they just select a higher deductible coverage for themselves, and levy that higher amount on the drivers, who don't take the waiver. In other words, the price we are paying for the CDW is only for the rental car company's deductible, not the cost of full coverage (i.e., like our 300K/100K liability plus collision plus comprehensive). I'm not sure about medical, but suspect there is none.
I take the standard CDW, and to date haven't had a problem. But, I have traveled with others in a large group who didn't, and the car rental company (not AB) went over the truck with a fine toothed comb, finding anything that wasn't documented at checkout, to charge them for. Usually, people are in a hurry to make their flights, and can't afford the time to challenge the potentially false claim. But, with the CDW, they just give the car a quick glance over. That's been my experience, YMMV.