Is it possible to restore the backups (web and database) that are created by the built-in backup feature under Site > Domain > Backup to a new Debian 12 server with a freshly installed ISPC? I tried adding the same website and then copying the backup files to the same location on new server but ISPC does not see them. Same version of ISPC and Debian on old server, just changing providers.
You should be able to use the files for restore but I think you need to restore by hand and can't through ISPC. Or maybe wait for a while for ISPC to pickup the copied files? Not sure if ISPC does that and how long it would take.
ISPConfig will not pick up the copied files on its own. Backups must exist in the database to be restored. So either you restore them manually (its just an archive of the site plus an archive of the SQL dump of the database that you can unpack and then copy / import) or you could try to manually add records for them to the dbispconfig database, but that would be quite tedious. In general, it would have been better and easier to e.g. use the Migration Tool to transfer your setup to a new server.