I hate to say it, but that argument doesn't quite hold up. Purchasing a gun is actually much more restricted than having a car. The only real exception is the statement about insurance, however since "gun insurance" does not even exist, it would be a little difficult to require it. If insurance companies would be required, by law, to offer such insurance, I suspect something like that might be able to be done.
As for other restriction points, the age limit to get a driver's license is 16 in most, if not all states. It is 18 or 21 (depending on the state and in some cases, the type of gun).
There is no background check to purchase or even to get a license to drive a car. There is a skills test, but nobody asks if you have a mental illness, or about your criminal history or calls your friends to see if they would be nervous about you being behind the wheel. While different states may handle them in different ways, Federal law requires that all retail gun sales include some form of background check. If the Universal Background checks being proposed in Congress go through this will bring the standards and requirements in all states to the same. Nothing like this has ever been enacted or even suggested for owning or driving a car.
Safe gun training courses (for all types of guns) are available in all states and are required in most. This includes, and is even more true, of handguns and the requirements to obtain a license to carry these.
There is no law restricting the owner of a car from loaning, selling or giving that car to any other person they choose. There are laws about who is allowed to drive it, but you can legally turn over possession of your car to anyone you choose. The driving part is on the person who has it. With guns you can be, and often are, legally forbidden from transferring a gun to another person, whether giving, selling or loaning, for many reasons. If any person who can not legally purchase a gun for themselves, it is illegal for you to purchase on for them or to transfer (give or sell) one to them.
The bottom line is that LEGAL gun owners already are under many more restrictions than car owners. What should be done first is to improve the ability to enforce those restrictions and laws. One of the biggest would be making it possible to actually find out about higher risk mental illnesses. At this point, unless a person has already done something to be "flagged" there is no way for a gun seller or the authorities (including the people performing background checks) to know if a person is at high risk for violence due to a mental illness. There are difficulties in enforcement of many of the gun ownership restrictions (just as there are difficulties in enforcing driving laws) because law enforcement is often not allowed to do the things or enforce the laws they are supposed to. We need to correct these issues and have the laws already on the books enforced before we start adding even more laws which probably won't be able to be enforced either.