The reason you don't want to appoint a Ginsberg to be Chief Justice is that the Chief Justice gets to sway which cases to hear. What was great about Rehnquist is that he vastly limited the number of cases the Supreme Court would hear and lengthened the duration of time spent on these cases. This limits the power of the judiciary to some degree and does a better job of maintaining the status quo that just blazing through case after case.
Other major factors:
1) If on the winning side, the chief justice decides who writes the majority opinion.
2) The Chief Justice presides over any impeachment hearings of the President or Vice President.
Thank you - much appreciated!