However, I don't see what significance the "chief justice" has. Last Feb., my students and I spent two hours with Justice Thomas, and he outlined how the Court does things, and he basically said the CJ has no special power, no extra influence, nothing. Just one vote.
I think it's mainly a role for public consumption, for occasional pronouncements about the state of the judiciary, etc. May not matter as much as people think, but still could be really bad to have the WRONG person in such a visible role.