Straw purchase has nothing to do with the eligibility of any participant in the transaction(s) to legally own a gun.
The Supreme Court decision in Abramski v United States makes it clear:
Holding: Regardless whether the actual buyer could have purchased the gun, a person who buys a gun on someone else’s behalf while falsely claiming that it is for himself makes a material misrepresentation punishable under 18 U.S.C. § 922(a)(6), which prohibits knowingly making false statements “with respect to any fact material to the lawfulness of a sale of a gun.”
What is a 'Straw Purchase'?
The Federal form 4473 has the question 11.a.
a. Are you the actual transferee/buyer of the firearm(s) listed on this form? Warning: You are not the actual buyer if you are acquiring the firearm(s) on behalf of another person. If you are not the actual buyer, the dealer cannot transfer the firearm(s) to you. (See Instructions for Question 11.a.) Exception: If you are picking up a repaired firearm(s) for another person, you are not required to answer 11.a. and may proceed to question 11.b.
and the instruction sheet explains
Question 11.a. Actual Transferee/Buyer: For purposes of this form, you are the actual transferee/buyer if you are purchasing the firearm for yourself or otherwise acquiring the firearm for yourself (e.g., redeeming the firearm from pawn/retrieving it from consignment, firearm raffle winner). You are also the actual transferee/buyer if you are legitimately purchasing the firearm as a gift for a third party. ACTUAL TRANSFEREE/BUYER EXAMPLES: Mr. Smith asks Mr. Jones to purchase a firearm for Mr. Smith. Mr. Smith gives Mr. Jones the money for the firearm. Mr. Jones is NOT THE ACTUAL TRANSFEREE/BUYER of the firearm and must answer “NO” to question 11.a. The licensee may not transfer the firearm to Mr. Jones. However, if Mr. Brown goes to buy a firearm with his own money to give to Mr. Black as a present, Mr. Brown is the actual transferee/buyer of the firearm and should answer “YES” to question 11.a. However, you may not transfer a firearm to any person you know or have reasonable cause to believe is prohibited under 18 U.S.C. § 922(g), (n), or (x). Please note: EXCEPTION: If you are picking up a repaired firearm(s) for another person, you are not required to answer 11.a. and may proceed to question 11.b.
The instructions on that form also say
I also understand that making any false oral or written statement, or exhibiting any false or misrepresented identification with respect to this transaction, is a crime punishable as a felony under Federal law, and may also violate State and/or local law.
Answering 'yes, I am the actual buyer' at question 11.a, when the transaction does not meet the description, can be charged as a Federal felony, by itself. The code is 18 USC 1001, in relevant part
(a) Except as otherwise provided in this section, whoever, in any matter within the jurisdiction of the executive, legislative, or judicial branch of the Government of the United States, knowingly and willfully - (1) falsifies, conceals, or covers up by any trick, scheme, or device a material fact; (2) makes any materially false, fictitious, or fraudulent statement or representation; or (3) makes or uses any false writing or document knowing the same to contain any materially false, fictitious, or fraudulent statement or entry; shall be fined under this title, imprisoned not more than 5 years or, if the offense involves international or domestic terrorism (as defined in section 2331), imprisoned not more than 8 years, or both.
Transferring a firearm to a person known to be ineligible is an additional Federal felony. Transferring a firearm to an eligible person, after a transaction where 11.a. was falsely answered 'yes', does not 'cure' the original felony.