Broder's reference to the power of the president's bully pulpit as the lever that will shift public opinion against the Democrats is just another example of his inability to grasp that the public turn against the Republicans in late 1995 and early 1996 was a reaction, on the merits, to Republican excesses, not the result of some inscrutable black magic Bill Clinton managed to pull off with a few press availabilities.
The more obvious flaw in Broder's reasoning stems from another bit of Washington myopia. What killed the Republicans on the government shutdown, in addition to the pure recklessness of the stunt, was that the government did shut down. National parks closed. Various government services and functions stopped operating. It had an immediate and direct effect on people's lives.
Most people really don't give a shit if the Senate is operating smoothly or not. They won't notice, they don't care. Shutting down business in the Senate is not shutting down the operation of the federal government. The fact that Broder and the Note's mystical "gang of 500" are unable to comprehend this (plus, as Marshall points out, their cheerleading of Newt's bold action back in the day) does mean it's time for them to put down their pens and go into a line of work which is less damaging to the rest of us.