I don't know about influence but the fact that he received his second nomination in 2010 tells you it was because Obama was happy with the quantitative easing. Obama could have pulled the plug on the easing by getting a supply side guy in 2010. That's why this is Obama's fault. The agency may have it's independence but the nomination process insures that the president will get his way.
Well, I don't think any Democrat would nominate a supply-sider. He didn't have to endorse QE as the only monetary policy tool at that point. And what would a supply-side Fed Chairman do? He can't lower taxes. Seems as if he would possibly sit on his hands and stare Congress down to do something.