You are correct. Each of the pre-gen characters does in fact have some background that ties into the Lost Mines story. For example, the dwarf cleric has a bond of "I have three cousins in the Rockseeker Clan who are friends and cherished clan members." The patron who figures prominently in the story is one of those Rockseekers.
If you use the basic rules to create characters, I'd look over the pre-gens and have each of your player characters steal one trait, ideal, bond, flaw, or personal goal from the pre-gens. Alternatively, you can look over the adventure and find connections that make sense for the characters you generate. One idea might be to look at the various societies involved (Harpers, Lord's Alliance, ...) and tie your new characters into one of those groups.
My players have really had a lot of fun when they discover how they are tied into the adventure. Losing this connection would, in my opinion, really lessen the adventure.