I don't see how a fixed term length for the court would give senators any additional power over judges? It wouldn't make it any easier to remove judges from office or anything whilst in post, but it would mean that appointments cycles were timed less randomly, which is one of the objectively weird things about the current system - appointments are basically partisan and dependent on the president, but it's really random how many picks a president gets. If a SC justice had a normal term of, say, 20 years, and they were retiring on a staggered system, you could basically guarantee that every presidential election roughly corresponded to getting one supreme court pick, rather than the current system where one party can just be "well we got lucky and got three picks in four years, so our judges might now control constitutional decision-making for the next thirty regardless of how people vote in the intervening time".
I think my point about requiring supermajorities for the appointments is a more important one though anyhow - ensuring judges had permanent bipartisan support would make a
massive difference to who could be put forwards, and would in turn make it a much less high-profile campaign issue and instead something senators had to deliberate on more properly.
Not that any of the above is going to happen, because for some reason the US hasn't invited me to redraft their Constitution yet