Comment 7 for bug 1962479

Applications can't do anything about the OS's audio APIs.

The best solution is replacing both PulseAudio and JACK with Pipewire. You can then configure Mixxx to use the JACK sound API in Mixxx's preferences and it will Just Work.

Unfortunately Debian does not make this convenient like Fedora and Arch where you simply replace the PulseAudio and JACK packages with Pipewire packages:
https://wiki.debian.org/PipeWire

If you would like this situation to improve, I'd suggest advocating for Debian to package Pipewire better so that doesn't require manual configuration, and furthermore switching to Pipewire by default