> Also making sure there are no secondary sales without background checks, which is supported by (I think?) the majority of Americans would help.

How would it help at all? They've already proven willing to break the law with the straw purchase.

> All of these would stop repeat offenders, and if I recall, the majority of straw sales are repeat offenders knowing they don't get in much or any trouble buying guns this way.

You don't know what you're talking about. It's a felony, and precludes you from purchasing or even possessing a firearm ever again. It's not possible to be a repeat offender.