What are the reasons to add an additional authorized driver for rental cars in the USA?

I suggest ‘…drivers… not added to the contract’ is the only point.

That anyone rented cars and let you drive is indisputable, but no more about reasons or rules than your adding drivers.

When you use your own US-based car insurance, as well as a credit card, exactly what cover does that give?

When you don’t use rental-company insurance, what exactly do you think you’re giving up?

The point of adding an extra driver is to be certain that the policy covers what happens when that person is driving.

The risk of ignoring that rule is that you will not be insured for anything.

Just to add one more point.

It would be nice to hear from someone with first hand experience about issues with drivers who were not added to the contract.

As explained in Is there a European website dedicated to reviews of how well each travel insurance pays out?, it’s hard to find stories of first hand experience dealing with tourist/travel insurance because such incidents are quite rare. Especially so if we narrow our focus to situations where:

1. There was an accident (rather than a mechanical breakdown of the car itself, which is far more common)
2. The accident involved serious damage beyond the size of the deductible (most accidents only involve minor scuffs in the parking lot)
3. There was an unauthorized driver behind the wheel
4. The authorized driver honestly reported who drove the car at the time of the accident

I’d estimate only 1 in 10k people have such experience and thus you’re unlikely to get a first-hand report from someone on Travel.SE. But in countries where the rule of law is respected by businesses and courts, it’s wise to look at your contract for guidance on what would happen, rather than relying on anecdotal reports. Hence the advice to always add other drivers to the contract or avoid letting them take the wheel.

You’ll want to take a look at the rental agreement. Chances are extremely good that it contains terms that look a lot like this (example from Avis for US rentals):

14. Prohibited Use of the Car. Certain uses of the car and other actions you or a driver may take, or fail to take, will violate the
    Rental Agreement. A VIOLATION OF THIS PARAGRAPH, WILL AUTOMATICALLY
    TERMINATE YOUR RENTAL AND IS AN EXCLUSION TO AND VOIDS ALL LIABILITY
    PROTECTION AND ANY OPTIONAL SERVICES THAT YOU HAVE ACCEPTED, INCLUDING
    BUT NOT LIMITED TO SUPPLEMENTAL LIABILITY INSURANCE, PERSONAL ACCIDENT
    INSURANCE, PERSONAL EFFECTS INSURANCE, ANY ROADSIDE ASSISTANCE PLAN,
    EMERGENCY SICKNESS PROTECTION AND LOSS DAMAGE WAIVER (LDW) OR PARTIAL
    DAMAGE WAIVER. IT ALSO MAKES YOU LIABLE TO US FOR ALL PENALTIES,
    FINES, FORFEITURES, LIENS AND RECOVERY AND STORAGE COSTS, INCLUDING
    ALL RELATED ATTORNEYS’ FEES, LEGAL EXPENSES, FEES AND COSTS THAT WE
    MAY INCUR. It is a violation of this Paragraph if any of the following
    occurs:

A. You use or permit the car to be used: 1) by anyone other than an
authorized driver, as defined in paragraph 5;

In short, if you allow an unauthorized driver to drive the rental car, your rental agreement will be automatically terminated, along with any insurance coverage involved in the rental.

What does that mean? Well you’re now driving a car without a valid rental agreement. If you get caught out (likely if you’re involved in an accident, though it’s not impossible for it to come up in the course of a police stop), that could potentially mean you’re violating state law by driving a car you have no right to use, which could get you banned from the rental car company if not in more serious legal trouble.

And there will likely be significant issues if you are involved in a collision. Does the unauthorized driver have auto insurance? (Even if your credit card insurance does cover the claim, most credit cards that offer rental car coverage only cover damage or theft of the rental car itself, not personal liability for injuries and property damage outside of the car.) Does their personal insurance cover them when they’re driving a car they’re explicitly not authorized to be driving? Probably not; standard insurance policy forms have an exclusion for this:

Loss to any "non-owned auto" when used by you or any "family member" without a reasonable belief that you or that "family member" are entitled to do so

In short, letting an unauthorized driver operate the rental car voids your rental agreement and risks leaving the driver uninsured. Neither of those are good situations to be in.

Some rental car companies will waive the additional driver fee for members of AAA, Costco, or USAA, and they may not require a fee for the primary renter’s spouse or business associates (as part of a corporate rental); check the specific rules of your rental company for details.

Adding an additional driver to a rental agreement has to be performed at the time of rental. This allows the rental agency to verify that each driver meets the minimum requirements for the rental agreement (has a valid license, meets age restrictions etc).

If you let someone drive your rental car that has not been verified by the rental agency, and something goes wrong, I would expect your insurance agent to contact the rental agency and ask "is there anything we should know about the person driving the car?", whereupon your insurance agent would not be very happy with the answer that they were not authorized to drive the rental car. This could lead at the very least increased premiums (because the insurance agency now knows you don’t play by the rules), or even having your claim denied – See below about the standard conditions when using Amex for insurance.

Of course you could lie about who was driving at the time of an incident – but that opens you up to a whole lot of risk from another direction (the people who have the power to put you in jail)

I rent cars using my Amex card, and get covered by insurance that way. I would expect that other insurance companies provide similar terms. The Amex policy (warning – PDF) says:

Who is covered

Benefits are available to Eligible Renters. An Eligible Renter means
a Card Member with an Eligible Card and their spouse or Domestic
Partner, and Authorized Drivers.

What are the requirements to be an Authorized Driver?

An Authorized Driver must be authorized on the Rental Agreement
between the Card Member and Rental Company to operate the Rental
Vehicle according to the terms of the Rental Agreement.

What is Not Covered

…

3. use of the Rental Vehicle in violation of the terms and conditions of the Rental Agreement (including Covered Events occurring when: a
   person other than an Authorized Driver was in possession or control of
   the Rental Vehicle; or driving the Rental Vehicle outside of the more
   restrictive of the Covered Territory or authorized rental territory);

So Amex will deny your claim if your friend, who is not on the agreement, drives the car and is involved accident (even if they were not at fault, are licensed, and meets the Rental Company’s agreement restrictions).