At least this Republican thinks it's a way better anti-poverty choice than raising the minimum wage, for many of the reasons already argued here.

As to any difficulty the eligible have in claiming it: How hard would it be to add another checkbox to the W-4 asking about other sources of income? If your total income including the job you're filling out witholding for is estimated to be less than X, it's your employer's responsibility to claim the EITC for you and add it to your paycheck. Perhaps grease the skids on this a bit by giving the employer some tax benefit for doing so.