I didn't intend this to be taken as a personal affront to people who don't use LinkedIn, so I should have used less inflammatory language. My apologies.
I think you're actually shooting yourself in the foot, and as someone else mentioned - you're actually missing out on the very best developers out there by taking the shortest, easiest path to recruiting.
Works in what sense? Does it reliably help you make the best hires? Are you sure? (I hypothesize that it does not, and you might not even know it.)
Yes, every hiring manager has lots of resumes to scan. But I wouldn't consider that a logical justification for establishing an arbitrary criterion that's likely to weed out some of the very best candidates. (As others have stated, there anecdotally appears to be a trend wherein the very best don't bother with LinkedIn.)