The change would expose employers to lawsuits from rejected applicants without appreciably improving the chances of those it is supposed to help. You can make something illegal without making it hard.
If the law is passed, a hiring manager who doesn't want to hire someone who is unemployed still won't want to hire someone who is unemployed -- and probably won't have much trouble coming up with "reasons" to reject them. The manager just won't be allowed to tell these candidates they are wasting their time.
Companies would also get a new incentive not to post openings at all -- relying instead on informal referrals to evade the prohibition. They would also get a nudge to bypass the entire U.S. system of employment law by moving abroad.
The supporters see this sort of "discrimination" as comparable to shunning people on the basis of race or sex. But the unemployed are not a group that has been victimized by age-old laws and customs based on false stereotypes and baseless fears. They have no history of hostility to overcome.
What they have to overcome is a dismal economy that generates no new jobs. In a boom period, impediments like this would be of no concern. The problem the unemployed face is not that they are excluded from some jobs, but that there is so little hiring going on.
When companies find their business growing, the unemployed will gain hope, and not before. Without a healthy economy, this measure won't be helpful. With a healthy economy, it won't be necessary.