Fixing The Court
This is a pretty terrible idea, based on the number of problems I can see in ten minutes contemplation. What if, say, the Republican Party is burned to the ground as justice and common sense dictate, and is replaced, say by two parties? How much of their time will the ten spend wrangling over what other five to appoint? Will they need to hire and supervise investigative staffs. This would also be a pretty major wound to the Constitution.
I have a proposal which, IMHO, is both better and simpler. Require all judges to be confirmed by a supermajority of the Senate, say 2/3 or maybe 3/5. This should ensure that no extremists get appointed, or at least that nobody without broad support gets appointed. I note that before the extreme politicization of the courts by Republicans, most Supreme Court justices were confirmed by substantial supermajorities.
I am not opposed to having justices serve for limited terms though, maybe ten or twelve years, or until they reached age 75.