In FiveThirtyEight's wrapup of their Kavanaugh hearing coverage, one writer said (emphasis mine):
Second, in this difficult and circular path, the fundamental process of confirmation to the Supreme Court may now be forever changed. With a high-leverage midterm election on the horizon, and a sitting chief justice who values the legitimacy of the court above all, I’m not exactly sure where that leaves us.
There were also one or two other comments during their live blog on similar lines. This confuses me, however, since as far as I know the chief justice is stuck with whoever the Senate confirms, so his (or her) opinion on the nominee and what they may do to the makeup or perceived legitimacy of the court is irrelevant.
Is there any "soft power" the chief justice has to influence either the nomination process or to reign in the tendencies of a new associate justice they disapprove of? Is there some constitutional option I forgot or law I'm not aware of that gives them a way to self-regulate the court (similar to how either house of Congress can vote to expel any member)?