There are valid arguments against term limits for judges, too.
I think that the terms don't actually matter much. The real problem with SCOTUS is that it represents extreme concentration of political power in our system that doesn't have any direct checks on it. Then you have stare decisis, which makes it so much harder to overturn decisions - so even indirect checks are limited. That's why its lack of accountability (e.g. term limits) is such a big problem. It's also why it was inevitable for it to become more partisan over time - having a partisan majority on the court is basically the equivalent of having nukes, and once one side starts moving in that direction, the other will inevitably follow. And now we're at the point where many people vote for president solely on the basis of what kind of judges they will appoint - and it's not even an irrational approach.
So I think that it's better to make the court less powerful, such that those appointments are not quite so important. One particular idea that I had is to abandon simple majorities as the way to decide matters on the court, and run it more like a jury - basically, the only way to declare something to be definitely unconstitutional, or definitely constitutional, should be by unanimous decision, or perhaps a strong supermajority (say 7 out of 9?). If a panel of people who are specifically chosen as legal experts cannot agree what the Constitution means with respect to something, I think the most sensible interpretation is that it's ambiguous - but then going with a simple majority would be very wrong.
Instead, if the judges cannot agree, this should automatically trigger a constitutional amendment process. Basically, have them all write opinions explaining why they cannot agree with their opponents on the court, and what changes to the Constitution would be necessary before they can agree. Submit all those changes as proposals for ratification, using the normal process, except that only one can be ratified, and that automatically rejects the other changes. If one of the amendments is ratified, that resolves the issue unambiguously and with an explicit rule for future cases like that. If none get ratified, then the lower court decision stands, but it does not set a precedent - the constitutional question remains open, and can be challenged again.
I would expect this to result in more unanimous or supermajority decisions with less extreme and more narrow effects, because the judges would be more likely to try to hammer out a compromise to cross the threshold.