First, I don't see where the Obama administration has zig-zagged its way between strategies. Second, a bipartisan bill was what the Obama administration wanted - but it was the fault of Republicans that this didn't work out. Republicans wanted the same old failed strategies of the past eight years - tax cuts and more tax cuts*. It takes two to tango; intransigent Republicans didn't want a bipartisan bill, they wanted a Republican bill passed by a Democratic President. The Obama administration had no choice but to face this reality, and accept it.
*Particularly, as Newt pointed out, cuts in Social Security and Medicare taxes, programs which the Republicans have wanted to gut forever.