First, as stated in the pinned post “If you have a problem, PLEASE read this first”,
Please always include a copy of an
nvidia-bug-report.log.gz
file, which can be generated with thenvidia-bug-report.sh
script
…otherwise it will be very difficult for anyone to help you.
Second, 50xx support was added only in the very latest 570.86.16, so make sure you are using this. This means that the ugly *.run file is currently the only option for you (as the apt repo for debian-12 is at 570.86.15).