That would be far less intrusive and far less invasive a procedure than violating the physical privacy of very many and in some cases every traveller, which is where this is already heading at several major airports.
Seriously, how would a background check work? What are you looking for? Ethnic background obviously would be the biggest failure (although still done, see above). Credit history? Police records (international? Good luck with that)?
Either you disclose what you are looking at and "the bad guys" can trick your check. Or you don't and you're back on the security by obscurity road, doing "arbitrary" (to the public) things, annoying probably just as much.
I think this idea is just as flawed as the current procedure. And I surely hope no one is starting something like that, since this whole security BS seems like a race of arms to me lately..