I am by no means an expert at the review process, but I have managed a few reviews, and I spent a fair amount of time doing some things that aren't mentioned in Eric's list. 1. Helping a new library author get a review started. While this isn't specifically listed in the role of the review manager, it can be helpful. 2. Helping to navigate some of the politics of the Boost community. It's not bad most of the time, but I've had to intervene more than once. In general, a review manager can make the fairly bureaucratic process less stressful and more pleasant for someone who hasn't already gone through the process several times. I suspect neither Eric nor Peter need this sort of help, but some others probably appreciate the help.
Why do we need a review manager at all?
Primarily to avoid any questions or doubts about whether a library should be accepted or not. The review manager supposedly takes everybody's feedback into account, but makes the ultimate yes/no decision, and is even free to buck popular opinion. It's a representative democracy, twice removed: we elect the review wizard, who elects the review managers, who elect the libraries. Which would make the review managers the electoral college. ;-)
Another key job of the review manager is to collect all the feedback and present a TODO list to the author of the new Boost library, but that's secondary.