Who would determine what judge was rotated into the decision? Consider the case where the alternates are Antonin Scalia, Sandra Day O'Connor, David Souter, John Paul Stephens, and Ruth Bader Ginsburg. It makes a big difference whether the chosen alternate is Scalia or Ginsburg. It makes some difference whether it's O'Connor or Stephens. Note that they were on opposite sides in both Austin v. Michigan Chamber of Commerce and Gonzales v. Raich.
With a jury, remember that it takes a unanimous vote. It is therefore unlikely that the alternate would be a member of a group that is partisan on the question of that particular case. To make it more unlikely, the presiding judge allows the parties to disqualify jurors for cause (and usually a limited number without cause). It's not strange for a juror pool to start with a hundred candidates or more that are whittled down to fifteen or so. Where would the Supreme Court get a hundred alternates? And who would "preside" as judge?
Note that it is the appeals courts that oversee that process for unfairness. One reason that a case could be sent back for a new trial is that the presiding judge could have show lack of impartiality in jury selection. The Supreme Court has no such review. They are the final review.
The current system already allows for alternates in its own way. The lower court decision is the fallback decision. And a later decision with a different set of justices can make a distinct choice on the undecided issue. That way the justices may be chosen in the normal manner.
It's also rather unlikely for a Supreme Court decision to be tied. It can happen but it seldom does. I know of no examples of a sick justice preventing a decision. They'd just reschedule arguments. It only happens in the case of recusal or after a justice dies until a replacement can be seated. And it's not like nothing happens in that case. There is a default decision available for that case. It's just that it doesn't set precedent for other cases or resolve circuit splits.
Contrast with jury trials. Those are the original trial. Absent the alternate, you may have eleven jurors for conviction or acquittal who are blocked from a decision by one missing juror. Without an alternate, the trial would be declared a mistrial and they'd just do it over again. Having an alternate or three saves selecting twelve new jurors and redoing the entire process. It should not change the decision one way or the other. However the alternate votes, it is unlikely to change from conviction to acquittal or vice versa. At worst, it might change a mistrial to a decision.
In those cases where it does change a mistrial to a decision, then chances are a new trial would have given the same decision. Because apparently one juror was blocking the other eleven. Without that specific juror, the decision was obvious. The chances of a second trial having the same issue are low.
The Supreme Court renders final decisions (or makes decisions final by refusing to overturn them). As such, its decisions need to be as grounded as possible. Creating a mechanism of alternates would increase its ability to come to a decision without making the decision any more infallible.
If this were really that important, then the correct way to fix it would be for a justice to choose her or his own replacement. Scalia might have picked Gorsuch. Then there would have been no delay in pick a replacement. Similarly, if Gorsuch were an alternate, then he could have replaced Scalia if Scalia felt the need to recuse himself. Meanwhile, Stephen Breyer might pick Merrick Garland as his alternate and replacement.
Of course, to make any such change would require a constitutional amendment. Those are difficult in such partisan times. And they weren't considered necessary earlier. The battles over nominations didn't use to be so partisan. Stephens and Souter may have been known as liberals on the court, but they were appointed by Republicans as were the swing votes, O'Connor and Anthony Kennedy.
If you're wondering how I made my five picks for alternates, I picked the current three living retired justices plus the two oldest justices active in 2015. I picked 2015 because Scalia makes a better example for this purpose than Kennedy, currently the second oldest judge.