Reforming SCOTUS

I have just sent this letter to my Congressional Representative, Ed Case of Hawaii District #1

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Dear Representative Case

Thanks for the good work you do. I have written to you many times about the tax treatment of expats. This time I write to you about the Supreme Court. Flipping one or two seats and getting radical policy shifts is no way to run a government at any level, from City Council to Federal Congress. Our way of voting is killing us.

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I'm sure many wiser and more astute legal minds have weighed in on this question, but here for your information is my suggestion for depolarizing and depoliticizing the Supreme Court. 
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What if the Presidential appointments of federal judges and Supreme Court Justices were replaced with a revolving mechanism to appoint term-limited Justices? 
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Not merely to expand the Court, but to expand it in a principled way. Otherwise the political polarization will never be resolved.
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I propose the following overhaul: 
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Retain the 13 Appellate Circuits as they are
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13 Justices, one from each Circuit, each to serve a 13-year term. 
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13 regional Bar Associations, each choosing its respective Circuit's nominee by a rank-ordered selection process
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One Circuit per year in strict rotation replaces its outgoing Justice with a fresh incoming Justice
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Each year the most senior Justice finishes his or her 13-year term as "Chief" Justice, a mostly ceremonial function, then is replaced on the Court by a freshly elected Justice from the same Circuit. The Court would never be composed of the same 13 Justices for more than one year, continually evolving.
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Polarizing presidential appointments are out. Senate confirmation not necessary, but the Senate may veto a nominee by a super-majority vote. 
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The operative concept is CHURN.
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• What could possibly go wrong and how could we ensure that it doesn't?

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My afterthoughts not included in the letter:

I would begin by finding out which Circuits are not currently represented on the Supreme Court. It may be more than four, but I would choose four of the unrepresented Circuits at random (literally, choose a white ball or a black ball, or choose a Circuit number and match it with an unrepresented Circuit).

These randomly chosen four Circuits would go first in filling out the Court to 13 members. Then, the most Senior Justice retires and (regardless of his or her actual circuit) Circuit #1 chooses the retiree's replacement. The next year, rinse and repeat with a new nominee from Circuit #2. After 13 years, the Court would be fully constituted according to the plan outlined above.

Selection by rank-ordered voting among each Circuit's Bar Association members will not GUARANTEE never elevating an extremist left or right wing nut, but it would tend to give common sense moderates an edge.

I think a Constitutional Amendment would not be required for this, only the will of Congress to codify it into law.