Instead of having job openings posted by those who don't want them found what if people posted willingness to work, perhaps in some sort of registry. That way a company would have to prove that none of the people willing to work are qualified. I'm sure many qualified people would be open to moving.
At the very least, if you want an H1-B, companies should be forced to post the secret jobs on a standardized, embarrassingly public database. Think MLS, but for jobs.
Yes, something like this would be great. You could tie the registry to both IRS and SSA databases ensuring a) the job hunters are real, and b) the jobs offered are (eventually) real. It would also be great to carve an exception into liability law and require employers to give feedback to workers about a rejection. I'm sure this leaves lots of room for malefactors on all sides, but it would handle the biggest flaws of the current system.