[Tom Servo]

All the ATF cares about is whether you are keeping the gun

I'd like to see a citation for that.

There are two code sections pertaining to this. 18 USC § 924(a)(1)(A) states:

whoever knowingly makes any false statement or representation with respect to the information required by this chapter to be kept in the records of a person licensed under this chapter or in applying for any license or exemption or relief from disability under the provisions of this chapter (...) shall be fined under this title, imprisoned not more than five years, or both.

18 U.S.C. § 922(a)(6) states:

It shall be unlawful (...) for any person in connection with the acquisition or attempted acquisition of any firearm or ammunition from a licensed importer, licensed manufacturer, licensed dealer, or licensed collector, knowingly to make any false or fictitious oral or written statement or to furnish or exhibit any false, fictitious, or misrepresented identification, intended or likely to deceive such importer, manufacturer, dealer, or collector with respect to any fact material to the lawfulness of the sale or other disposition of such firearm or ammunition under the provisions of this chapter

It comes down to the act of making false statements. Moving on,

It does not matter who paid for the gun.

Untrue. On page 165 of the Federal Firearms Regulations Reference Guide:

The straw purchaser violates Federal law by making false statements on Form 4473 to the licensee with respect to the identity of the actual purchaser of the firearm, as well as the actual purchaser's residence address and date of birth. The actual purchaser who utilized the straw purchaser to acquire a firearm has unlawfully aided and abetted or caused the making of the false statements. The licensee selling the firearm under these circumstances also violates Federal law if the licensee is aware of the false statements on the form. It is immaterial that the actual purchaser and the straw purchaser are residents of the State in which the licensee's business premises is located, are not prohibited from receiving or possessing firearms, and could have lawfully purchased firearms from the licensee.

An example of an illegal straw purchase is as follows: Mr. Smith asks Mr. Jones to purchase a firearm for Mr. Smith. Mr. Smith gives Mr. Jones the money for the firearm. If Mr. Jones fills out Form 4473, he violates the law by falsely stating that he is the actual buyer of the firearm.

In short, if Mr. Smith's money paid for the gun, Mr. Jones makes a false statement when he says he's the actual buyer. If Mr. Jones uses his own money to pay for the gun, with the intention of giving it to Mr. Smith as a gift, he can truthfully answer that he's the actual buyer.