At the risk of being pedantic, what would it even mean to depoliticize the Supreme Court? The institution is inherently political because its judgments have implications for public policy. It's just not realistic to expect it to remain free from partisan & ideological concerns.
The best you can do is to have an informed, active, & organized public that makes its voting decisions with a relatively impartial judiciary in mind, &:
Maybe not depoliticize, but do what we can do depoliticianize
Oh, then we can't. Republicans wouldn't get behind anything that would erode their ability to stack the courts, including your proposal.
