I agree that criminals don't obey laws , thats why I carry. They will likely purchase their illegal weapons from an illegal dealer. The law has no point if it won't stop the criminal from getting a gun.
However, if we make private sales require a background check, that limits the market for criminals to buy their weapons and the cost of an illegal hand gun goes up. Thus, making it more difficult for the typical convicted felon to acquire one. Also, you must consider the fact that without a background check, the well intentioned private seller...as I assume you are referring to when discussing inconvenience issues.....cannot be sure that he is not selling a deadly weapon to a criminal who intends to use it for his nefarious deeds. As an individual, it is not my responsibility to ensure another is not breaking the law. Despots use ordinary citizens to "keep each other in line."
Even if the a private seller wanted to conduct a background check to ensure he is not selling his weapon to a criminal, he would be at a severe disadvantage in competition with others who don't care who they put a gun into the hand of. If you had a law requiring private sales to use back ground checks, that would put the well intended seller on the same level with other sellers. Again, the private seller should not have to worry about putting a gun in a criminals hand. We are not the police, and I don't want private citizens being required to act as such. Despots use citizens for police.
As to the size of the inconvenience, I would respectfully argue that the prevention of a criminal from acquiring a firearm out weighs the inconvenience of a 3-5 minute phone call. Maybe YOUR 3-5 minutes, but not mine. Keep your hands off my time and freedom. BTW, it takes 3-5 minutes (twenty, the last time I bought a firearm) for a dealer. An individual selling his .22 rifle is going to have to figure out the process and risk breaking some obscure law or pay someone to handle the process for him. The net effect of require checks for private sales will be the virtual elimination of private sales. Quite probably, this is the real goal.