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This is completely backwards - the question is not what will the Republicans want to vote for, but what will they have to vote for. With a handful of exceptions, when Republicans sought Democratic votes during the Bush years, they did not do so by crafting broad bipartisan compromises; rather, they created legislative environments in which certain Democratic Members felt that, from a political perspective, they had to vote for Republican proposals. Even with the shrunken Republican Congressional caucuses, there are still numerous Republicans sitting in swing seats who can be made to feel political heat if they vote straight party line.
Like your argument regarding "up or down votes," I don't believe that adopting this approach would necessarily translate to immediate legislative success - Republican party discipline is high at the moment and the political wind is at their back. But, as you note, in the present environment, the Democrats have the worst of both worlds - Republicans get to obstruct their political agenda and pay no political price for doing so. Switching to an approach that focuses on putting political pressure on Republicans to vote for Democratic initiatives at least has the potential to bear fruit (legislatively or electorally) down the road.