You mentioned you have restored from a backup. You should check to make sure you restored 2 things: files and database.
Some backup plugins may not offer full backup, it could be only files or only database. Your case sounds you're not restoring full complete backup. Try restore again but both files and database.
Never have experience like yours. But I ever made experiment, deleting database of a working website will cause WordPress showing the install screen asking you to install as a new website. So I guess, your restoring was not complete, some data is not matching for example your wp-config.php telling the database is pointing to something not exist.
If you have no files backup to restore, what should you do? Try to examine your wp-config.php, edit it and make sure all the settings inside the file are correct.