They hate writing checks just to write checks.
You can get it passed if you sell it as a progressive subsistence voucher that will replace the minimum wage, medicaid (we still have an insurance mandate), food stamps, etc.
In fact, use the word voucher. The GOP loves those.
...and the GOP is more complex than people understand. I voted for Romney and McCain and I'd go for an EITC if it came with the right reform package. All this "if it weren't for those guys..." is really counterproductive.
Over the last 30 years the GOP has raised government spending far more than the Dems. Consider the Medicare prescription drug plan without any negotiation for lower rates.
It's just redicusly over the top levels of corruption.
Can you name a political party, in any part of the world, in the last century that did follow through with their "great message"?
Politicians are PR / sales people. The product work is rarely actually done by them.
If mortgage interest rates on average went up, the politicians will blame the greed of banks, the international exchanges market, the gravity was higher than normal etc.
If mortgage interest rates on average went down, then that's a great reason to vote that politician in for another term!
People who had suffered through WW2 wanted to feel that there would be some benefit to the common people - 70 years later and the NHS is still one of the most loved institutions in the UK despite being avowedly a socialist endeavour.
NB I am not a supporter of the current Labour Party (AKA "New Labour") but they do deserve a lot of credit for what they did back then.
The nearest example I can see in recent history was the ANC in South Africa since 1994.
The problems with the suggestions here is #1 and #2 are both vigorously defended by different parties. So your proposal is a no-go politically. The Dems won't lower or get rid of minimum wage, and the earned income tax credit is already hated by the GOP (even though they proposed it). There is the additional problem of all the missing details as well.
Now before anyone trots out the whole "well if politics werent so stupid..." I have to stop you there. You reform with the politics you have, not the politics you want.