I have searched around, but can't find any "default" setting for the nametag. How can I make all new players on my server receive the nametag color by default?

You need to order your groups.

If you're using a flat file system (groups.yml), Make sure the groups are from the highest to the lowest.

If you're using a MySQL system. Use the order command to make the groups from highest to lowest in-game.

/ne groups order Owner Admin Default

Once completed, make sure you give groups permissions.