Where have all the illegal guns in the hands of criminals come from?
Most are stolen from individuals, or acquired through straw purchases. Neither of those things are stopped by the NICS system.
NICS was an unpleasant compromise we accepted back in the 1990's, and I see absolutely no reason to expand its reach. Ask someone who's been wrongfully denied what the appeal process is like.
Oh, and as far as guns "stolen" from FFL's: most of those are book-keeping errors, sometimes on the dealer's side, sometimes on the distributor's side. Any time there's some question as to acquisition/disposition and a trace request is issued, the gun is a crime gun
. There is no other classification in the system.