LSU is still the school of Bob Pettit, Pistol Pete, Shaq, Chris Jackson, Big Baby etc. They've also been to two FFs since STJ's last. They ought to be able to attract bigger names.
Being 2nd fiddle at Fla hasn't forced Billy D out.
The problem with your hypothesis is that you assume Billy D was Billy D before Florida. Uh he was at Marshall for what 2-3 years with nothing major. He did not get to be Billy D UNTIL he made at Florida.
If LSU finds their Billy D fine, chances are though it will be someone who is NOT established yet at least at the BCS level.