Save As doesn't break relationships -- it's the same save with a new name. You can delete the older save the next time you go into the game. Using a number as an identifier helps; you can also check the last saved date to make sure you're deleting the right one.
Another good use for Save As is if you want to set up a template file with new houses/venues then use that as your jumping off point. Just use Save As right after you start the new game, and that keeps your template in pristine condition for the next new save you want.
The one drawback with Save As is that you don't have an older save to roll back to if something goes wrong and you need to backtrack, once you've deleted the older save.