It's unfortunate but having a stalemate is the best course to eliminate rash changes to government when there is a lot of divisiveness.
The big difficult, however, is how to make sure that a hostile senate wouldn't just block all nominations (which is the new norm). Some people like Calabresi have suggested that the president and everyone in the senate would be denied all compensation (IIRC even from private parties) until the seat is filled. I don't think that's a workable idea though.
It's an appointment, so they would only have to convince the president. Or you can prevent reappointment at all.
Most presidents are reelected to two terms, and so if judicial terms were offset from Presidential terms by a year, they'd have:
At 0 years, 0 justices they appointed.
At 1 year, 3 justices.
At 3 years, 6 justices.
At 5-8 years (and up until 1 year into the next term), all 9 justices.
So most President get to completely stack the court, the effects just don't last as long—the judiciary becomes another weathervane political branch of government, because the two of those was too few.
(And I don't even want to consider the consequences of all those short time justices looking to their post-Court career while sitting on the bench.)
That's how the Senate used to work, with state legislatures "electing" the senators.
That was replaced a hundred years ago with direct election of senators.