While all of the above is fine and will work, it means that your library files will be on an external drive, which is going to be slower than an internal drive, in most cases. Especially if you have a fast SSD in the PC as a boot drive, and that is where the library has resided until now.
My advice in your situation would be to leave the original library where it is, and just change the location that MC backs up the library to, using the setting at Option>File Location>Program files>Library Backups. That way, while the library itself isn't backed up to Google Drive, all backups of the library are backed up to Google drive. As the backups happen automatically on a regular basis, this will give you a backup almost as recent as if you were directly backing up the library.
Also, if you have Google Drive doing incremental backups near real time (I'm not sure it can be set up this way, but OneDrive can, so I assume it can), every time a file changes, then the Google Drive backup process is going to be very busy, and may interfere with MC, as it locks files to do the backup. If you have Google Drive set up to just do a copy once a day, this would probably not be an issue.
TL;DR. Change the location of the MC library backup to your external drive, not the location of the library itself.