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?
Because the Court needs to be larger, not smaller than it is today. Just to keep on average nine justices at one per term would require an average term of 36 years and only one Justice so far has made it that long on the Court. Ideally, there would be thirteen, one per court of appeals.