Your friends are correct in that you should submit at least one rec from someone who has supervised your work. Not doing so can be seen as a red flag. When I was on the adcom, recs were always seen as something that could not get you in, but something that could def keep you out. a bad, generic, or incomplete rec was always a red flag. and an excellent rec (for a top program) is often table stakes.
with all that said, the adcom cares less about the 'level' or title of the person providing your rec and more about the substance. with that in mind, were there others you worked closely with who might not have been your direct supervisor? are you able to complete any consulting assignments between now and when you apply?
you should bear in mind too that for re-applicants, the adcom will still read your original application along with your new one. they will see your recs from last time, so they'll know about the duplicate rec issue, and may not factor it as heavily for the reasons you've already cited.
hope that's helpful,
-james young