Suppose there are multiple such systems, each with different sets of users. Someone's infected while at a train station, and only 25% of the people are contacted. Should those who are not contacted feel safe? Or extra worried, compared to how they would have reacted for having no such system?

As I wrote: "What benefit is there to participate? Wouldn't the majority of those people prefer to enable location tracking on their phone instead of manual QR checks? "

If most would prefer automatic GPS tracking over manual QR checks, the what you're proposing would result in multiple systems and incomplete coverage both for users and for people who would set up the QR codes.