That is exactly what is now on the cards. Our gun laws are not national although in each state they are fairly similar. The national government has control on which guns are prohibited imports that can be seized by customs.
The loophole is that one reason you are allowed to own a firearm is being a member of a gun club. Certain classes of firearms are prohibited. The older man, the father, was a member of a gun club and had six long guns. The Prime Minister is talking about restricting the number of guns allowed but I cannot see that doing very much. He is also talking about periodic reassessment of fitness to hold a gun licence. That does not apply now.
I have to subject myself to applying for a fitness to work with children check every so many years. My name is then checked against a national register of people whose behaviour has raised red flags. If I have come before the courts for certain offences my name would be on the register, and I would not be allowed anywhere near children. This applies to paid workers like nurses and also to people like me who volunteer at a preschool, a school, or who are youth leaders. Scout leaders, music teachers and anyone else who has access and contact with children must submit to periodic scrutiny
It makes sense to apply similar checks for gun ownership.