ah - I see where you are coming from.
When a gun is involved in a crime, you want the perp to be punished for purchasing the gun illegally. But you you don't think the person who sold it to him illegally should be punished.
There are several ways to achieve the trace back to the seller. A sales database, testimony ...
You did not answer the question:
How does requiring me to run a background check on the guy in the parking lot prevent me from buying the gun and handing it off to him w/o running that check?
The answer to this question goes directly to supporting, or destroying your statement that "If EVERY gun sale required a background check, criminals would not be able to use straw buyers."
Further:
You agree that background checks are a form of prior restraint, and that standard for upholding a prior restraint is a very high one -but, even so, prior restraint has been upoheld by the court.
Specifically, on what basis? That is, when did they do so, why did they do it?
Then, explain how this example can be soundly applied to the issue at hand - background checks.