Even if a poll response represents good polling practices, and this one might qualify, it's tricky to translate that directly into a budget deal. As you yourself have pointed out, although some polls show that budget cuts should be the #1 priority, nobody, including the GOP, can find any significant cuts that are popular.
If polls asked "Would you be in favor of raising taxes by eliminating deductions X, Y, and Z," then "raising taxes" in that sense might not be so popular. Maybe that is what the GOP is trying to get at, but I doubt that would work very well, since, for one thing, polls don't ask the question like that.
I can only guess that they are trying to get O to go along with future cuts in medicare because they believe a Republican POTUS could never do it,
And they don't like raising taxes and figure the worse the deficit gets the more appealing raising taxes will look.
So they let O spend now, they make a compromise deal on taxes to get cuts in our future entitlements. Is that the plan????
I see no reason to believe that Bohner will beat O with his strategy, whatever that is,
and I am not thrilled about me going on O care at 65 (if I want to retire then) in return for continuing spending now and a deal on taxes now, and Obama looking like a big bi-partision success with Bohners help.
Just to be painfully repetitive:
But what if Republicans send two bills to the Senate, one extending cuts <$250K (for everyone) and one >$250K ?
Then they are not :
1) Raising taxes, or
2) Holding the middle class hostage to the rich
Then its Dems that raised those taxes.