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With broad public support, Obama and the Congress will at least have a shot at getting enough semi-moderate Republicans and conservative Democrats to vote with him on large issues such as the economy, health care, energy independence and the environment.
I don't completely follow. Obama already has broad public support. His enemies will spend the next 4 years chipping away at it, and it will slowly dwindle, pretty much no matter what he does (though he can do many things to either exacerbate or mitigate that process). Furthermore, unless the Democrats somehow achieve the impossible they won't have 60 votes in the senate, so compromise with the disciplined, and as it dwindles ever-rightward-drifting, GOP minority is going to be necessary pretty much all the time, whether or not Lieberman is chair of Homeland Security.
As for support for specific Senators, neither Reid nor Lieberman are very popular at home right now. How is hitching the Obama wagon to those horses going to help in any way?