Missing mysql, sys, performance_schema
If you see the slow log and general log in /var/lib/mysql/mysql, it means that in the config files for the old server, these logs were specified without directories, and that in the old server, the datadir is actually configured to be /var/lib/mysql/mysql, and not the default /var/lib/mysql. That would explain the appearance of the logs there. Check the old servers .cnf file and make sure the datadir matches before attempting the restore again.
After you figure out where the correct datadir was on the old server, match that in the new server, and do the restore. Be sure to set ownership after restoring: &quot;chown -R mysql:mysql /var/lib/mysql&quot; (or wherever the datadir ended up)
Also, keep in mind that mysql_upgrade or update do not recreate much from scratch. They mostly modify system objects to match the new version, and leave others intact, so if something is outright missing (like an entire system schema) it probably won't work.