The BATF can walk into his store at any time and demand to see his bound book, and god help him if his inventory doesn't match up with his records. That's how businesses get regulated.
Bob and Alice have been married for twenty glorious years. Bob wants to take up target shooting so Alice decides to give him a gun for his birthday. Alice goes to the store with the make and model of the gun he wants, fills out the paperwork pays for the gun and leaves. She gives him his present over dinner that evening.
Bob and Alice have been living together for five years. The neighborhood is going downhill and they can't afford to move, so they decide to get a gun. Since Bob is a truck driver and is often away from home, he tells Alice what to buy and she makes the purchase.
Bob and Alice hooked up at a rave and fell madly in love. Bob thinks guns are cool so Alice goes to the store and buys him one.
Bob and Alice broke up because Alice found out about Cathy. She kicked Bob's cheating ass out but kept the gun. Alice meets John and falls madly in love with him and gives him the gun as a gift.
I could write scenarios like that all night. There is no way the guy selling the gun to Alice will know what kind of relationship she has with Bob or anybody else. There is almost no way to distinguish a legitimate relationship between two people and a straw purchase at the point of sale.
Now, if you want to require background checks on firearms transfers between private parties, you have to record chain of custody. Which is to say, you have to turn every firearm owner into a firearms dealer with all the rights and responsibilities thereof. Even if you make access to the NICS system available to private citizens, anybody who wants to circumvent the system will do so unless you attach some means of verification and penalties for non compliance. Otherwise when the cops come knocking people will just say "I gave it away/there was a tragic boating accident/it was stolen/I dunno what happened to it" or whatever other lie they can think up and nobody will be able to prove otherwise or prosecute them for selling the gun to the wrong person.
Remember the bound book? If you turn every firearm owner into an FFL you will have to allow the BATF to have access to their private residence at any time. The government can walk into your home at any time and demand to see your records. That idea will go over in this country like a turd in a punchbowl. Any political party responsible for that law will get a one way bus ticket to the political wilderness.
Corrections and amplifications of how the firearms distribution systems works are warmly solicited.