Obama runs things. The Obama-Pelosi “tug of war” is just a good cop/bad cop riff to fool the public. Obama is above the fray. Anything that goes wrong is someone else’s fault. The good parts of the stimulus bill belong to Obama, the “bad parts” to Pelosi.
I think you are exactly right.
And the Republicans are too cowed by Obama’s approval ratings to take him on, directly, so they direct their criticisms at the legislation and Pelosi, instead of taking Obama to task for his lack of leadership.