Think about it this way: Should a car dealership be required to check the driving record of someone buying a car? It's wrong to imply that the seller should somehow be responsible for the actions of the buyer (too close to Bloomers view). How about the fact that a felon would be breaking the law by having a firearm and an illegal shouldn't be here anyway? At some point, people should be responsible for their own actions.
No, but try taking a car dealer's demo for a ride without showing your driver's license. So, essentially the dealership is checking your record because, if you had a bad one, you wouldn't have a license would you?
No one has noted that in general this should be a state rather than a federal issue. The only federal role should revolve around protecting rights.
Actually sellers are responsible for their actions. If someone knowingly sells a firearm to a felon, they'll be held civilly liable, and in many (most?) state have violated the law themselves. In Illinois, which has some of the most restrictive laws, a private seller is essentially held to the same standards as a dealer, except for the check. The seller and buyer must have a FOID and a record including the serial number must be created and retained. There's no need for additional laws, particularly at the federal level.