Anyone who believes judicial elections, be they by the legislature or by the people would cure the ills bedeviling the court need only look to the States to see the folly of that belief. That said, there is no constitutional requirement that SCOTUS have a fixed sized. Having the the President nominate a new justice at the start of each Congress and not bother with nominating "replacements" when they die or retire would be a way to prevent the lame duck nomination fiasco we now face and it wouldn't require a Constitutional amendment.
But Congress changes every two years, that's two Justices per term.
Why not just one justice at the beginning of the presidency, with no replacements and no re-election appointments?
