If everyone says to themselves "Well, I won't rent out my car, but I'd rent someone else's" then there will be no cars to rent.
There has to be people willing to rent out their cars for this to work, and I think most people would agree that renting out your car to strangers is a terrible idea.
And don't think for a minute that insurance companies won't raise rates if this takes off, despite what any law says. Someone's insurance company has to cover the car, whether it's the person renting the car or the owner of the car. If owner's insurance covers, then they're going to raise rates because my rates are based on my driving history, my age, etc. If I'm letting random people with varied driving histories and ages drive my car, they're going to raise my rates to cover the increased risk. If renter's insurance covers, my rates are based off the kind of car I drive. What if someone normally drives an Aveo and pays rates on an Aveo, but then goes out and rents somebody's BMW M5? Again, insurance companies will need to charge more to cover that risk. If you're normally paying for insurance on a $10,000 car but can easily go out and rent a $90,000 car, they'll charge more.
The simplest reply is if nothing else it transforms car rental business from one only for large corporations to one of those home businesses that everyone romanticizes with the lowest startup cost of almost any business short of standing on the corner selling candy.