I have fixed and restored a client's broken zimbra's mailbox database. It was the exact same problem as yours. Zimbra installation couldn't start, because MariaDB couldn't start. In this case, the problem was not from uninstalled MariaDB, but a broken transaction log, without any backup. Thus MariaDB refused to start. I've succesfully restored and reinsert all missing messages into MariaDB, then recover Zimbra. Zimbra is a mail server, with all messages are indexed into the MariaDB database.
If you have the problem with Maria DB installation, it should be very easy for me. If all else fails, I also have experience installing MariaDB from scratch, manually, by compiling from its source, to fix some incompatibilities with the installed libraries. So if you can't reinstall MariaDB by package, I could install it by compiling the database from source.
If you wish, after the plesk recovered, database and plesk setting backed up properly, I could remove everything, fresh install it, then restore the data back. So you can start clean.
I have fixed similar problem in the past, with higher complexity. So, I know exactly what you need, what I have to do to fix yours.