Hopefully it would give some needed weight to arguments against lifetime appointments.

This is a really really bad idea. The whole point of a lifetime appointment is that the Supreme Court should be "above the fray" of 2-year political cycles and the general horse-race nature of American politics. Just turning it into another position dependent on which way the political wind is blowing at any given moment, like the House/Senate/Presidency, would denigrate the role of a Supreme Court Justice.

Now, one could argue the court is already politicized, but term-limiting would make it 100x worse. Right now everyone gets whipped up in a frenzy whenever a Justice retires or dies, which is pretty infrequent. Imagine the Supreme Court being in play during every election cycle.