Sure there would be the issue where the outcome of your case still depends on which particular group of judges hears is. But it's not much different from what we have now where the outcome depends on which particular judge dies and which replaces him.
Just change it so that whichever particular group gets the case, the have final say and it applies throughout the nation.
The problem with that is the cases are representative. Let's say there are two Supreme Courts. Through whatever fluke of timing, One of them gets packed during Reagan's terms, and the other during Clinton's terms.
So the Conservative Court A takes an abortion case, and somehow finagles overturning Roe. Abortion is now illegal. At the same time, Liberal Court B is doing the same thing to Heller. So while Court A takes a case to re-establish Heller, Court B re-establishes Roe, then expands Fair Housing to cover Purple People from Planet Xylon. Conservative Court A gets a little miffed, and not only gets rid of protection for the Purple People from Planet Xylon, they strike the Fair Housing Act altogether for discriminating against people not in a protected class at all.
And they continue back and forth. Not only does everyone have to keep track of both courts to know what the supreme law of the land is, it's flipping back and forth as the two courts bicker with each other. So you may have had a right to concealed carry when you applied for your permit but by the time the State gets around to issuing it, you don't. So they refuse. Then they have to issue it, and the day it shows up in the mail, they no longer have to issue it, so they revoke it that day.
Even worse, what happens if they take the same case? People for Superpacs sues the FEC to be able to donate more to political campaigns gets taken up by one court, while People for Individual Expression suing the FEC because they think ANY money from superpacs are destroying the one-man-one-vote concept. Then on the same day they issue opposing rulings. Then what?