How big are these companies, how many people do they hire per week, and how many do they interview? Any number of perfectly reasonable interviewing strategies in smaller companies fall apart at big-company scale (1000+ resume submissions per DAY)
If you're getting 1000 resumes per day, I think you would be able to weed out a significant number of them just by giving them a coding assignment to work on. Churning out a resume is easy, but sitting down for a few hours and writing well-designed code takes effort.
How would you grade/score ~1000 code submissions per day, though? You could conceivably do Coursera/Topcoder-type automated grading, but that can only get you so far and can't distinguish good code vs. bad code vs. "copied from Glassdoor" code. It might be useful as an initial filter, though. You'd have to constantly be implementing new questions w/ associated grading scripts, though, as the problems would inevitably leak.