Note your "old board" is more than just the database. The database contains all the messages, posts and configuration options. Any files/attachments that had been uploaded, and any avatars or other forum images that were in use, are part of the file system and not the database.
So if you're truly intending to "upgrade the 3.0.12 board we had running before", you likely need to be doing FTP uploads of the /images and /files directories from the 3.0.12 board, in addition to restoring the 3.0.12 database. So that the 3.2.3 upgrade process has access to all the files being referenced by the database. Unless you just happened to be in a case of "the old board had zero attachments, zero avatars, etc."
As to your original issue of "entering the /install URL just displays the folder contents", that sounds like a case of entering "ftp://mydomain.com/install
" instead of "http://mydomain.com/install
". The "ftp://" prefix is asking for an FTP connection instead of the web server, even though the paths are otherwise identical. To start the phpBB install/upgrade process, you need to use "http://mydomain.com/install