Those would be a good start.
I got some other smart suggestions from a socially liberal gun-owning friend (listen up, Mr. Biden): Licensing - with passing a safety course as a requirement; and registration.
This friend suggested making unsafe handling of a firearm a felony. Those found guilty wouldn't necessarily go to jail or even pay a fine, but they would be banned from gun ownership for some period of time (life if egregious). You would lose your license and any firearms you own.
Gun registration would prevent a lot of straw purchases. Unregistered guns could be confiscated as illegal.
To me this makes sense. For the most part, it's how we treat car ownership and operation. You cannot operate a motor vehicle without a license - which is available only after passing a test. Also, all cars must be registered. And reckless driving has its legal consequences.
Of course, the radical gun nuts would probably try to point out that there's no constitutional amendment guaranteeing the right to own or operate a car as there is to "bear arms".
As my friend then said, "I'm a firm believer in the right to own guns, but I'm also a firm believer in people having to take responsibility for those guns. That's the part that I find lacking on the right."
Indeed, with rights come responsibilities.