A “QR code system” is just a method of pointing you to a website. That’s it. What SHOULD be happening is that website should link to a government database that the contact tracers access when dealing with a cluster.
That would be ideal, but for the moment government contact tracers (in Qld at least) can access the QR code data in a relatively quick timeframe. They contact the club, shop, pub, restaurant or any venue and request the attendance data, which all venues were required, under their 'Covidsafe' plan/authority, to make available within hours. The venue then advise their QR code provider who sends the relevant data to the contact tracers in digital form.
Yes, it could be done in fewer steps as you say, but it's better than a pen & paper record.