QR scanner on machine reads your unique QR from your app to log you into the machine, not many other considerations, 1 to 1.
The other options would have more fail points, considerations, development efforts for what is supposed to be the simple part of the solution and allow them to spend more time developing new features/releases for the app.
What if multiple people are connected to the machine on location with their phones? Don't need to worry about network related connection issues, handling of additional logic to have to go back and forth between devices a few times to validate the appropriate user that is playing.