President Obama campaigned as somebody who would transcend partisan politics. At a gut level that probably was his inclination.
However, being inexperienced, he had to rely on people who did not share that inclination: Harry Reid and Nancy Pelosi. This became apparent in the first stimulus bill, which no House Republicans and only three Senate Republicans backed as there was no GOP involvement. To a lot of people, Obama’s promises became lies.
Reid and Pelosi said that the GOP was obstructionist and wouldn’t cooperate. Plus with the supermajorities they weren’t necessary anyway. On the one hand, there definitely would be a lot of Republicans who only had the aim of being obstructionist. On the other hand, Obama in 2009 was an very imposing figure - he had a massively high approval rating and had won a great many states and house districts which had Republican Senators and Republican House Members ... those people probably would have happily involved themselves in the process of drafting the stimulus bill. Instead a bill drafted by democratic leadership behind closed doors was put on the house floor - this set the tone for the Republican relationship with Obama going forwards.
Republicans figured that a stimulus would involve explicit tax cuts, more money for shovel-ready infrastructure jobs, and more bailing out of state governments than then OTL bill had.
What if Republicans had been involved in the drafting process for the stimulus and the tone for the Obama Years was began with a big bipartisan stimulus bill? I suppose this would require the Democrats not to have supermajorities - which could be partially resolved via Coleman beating Franken. Alternatively, Obama could be more forceful with Reid and Pelosi on the need for bipartisanship.