I don't think you know much about gun sales. It takes like 2 weeks to purchase a gun because you have the legally able to purchase a gun, the dealer has to run your information, and the serial number has to be associated with your identity with the ATF. Some places even do ballistic fingerprinting making it so that if a bullet is found and intact enough the precise gun that fired it can be identified with some margin of error.
Also mental illness is already taken into account when you want to purchase a fire arm. There's really a lot to the whole gun purchasing process in the United States that people who have never even attempted to purchase a gun before like to complain about.
It also doesn't take a background check to buy a car. EVERY gun sale requires a background check from the NICS and has for an eternity.
And just to be clear, I don't own a gun myself, but I have friends who worked in a gun shop, who did offer me a handgun at a great discount, but the amount of effort it actually would take for me to purchase it was seriously not worth the time for me to do target shooting every once in a blue moon.