The problem with referenda that the function of the Supreme Court isn't writing new laws but settling individual cases according to existing law. The reason why this has wide reaching implications is because other courts are bound to follow its legal arguments as precedents with other cases (and because Supreme Court justices can sometimes rule that a particular law is inconsistent with the constitution, which they do at least in theory with the intent of preserving the constitution rather than creating a new law)
The public is clearly not bound to even attempt to be consistent in its opinions from case to case, and if it's all over the place in its verdicts and might well be accepting or rejecting a verdicts for different reasons from the justices (maybe they think the legal arguments about privacy in the original Roe vs Wade were weak but also think women should have an inherent right to abortion that legislators haven't provided them with), it'ś difficult to see how lower courts can be bound by it.
There's also a big question of who gets to vote, particularly for the cases where a state law is argued to be in violation of the federal constitution, which tends to be the more controversial ones...
The problem is that existing law in many cases (that end up in SC) is simply too vague to be a meaningful expression of intent. In those cases, since the judges are still required to rule one way or another, we end up getting those tortured decisions where a big effort is made to contort the law into some interpretation that would allow for a ruling.
In my opinion, if SCOTUS is split 5/4 (or even 6/3) on anything, that should be taken as prima facie evidence that the law is unclear and needs amending. At that point we should require all the deciding judges to make a write-up for each opinion in the case detailing the requisite changes to the law that would make all judges to rule unanimously in favor of that opinion. These can then be submitted to the legislature or a referendum or whatever, but either way some decision needs to be made.
The public is clearly not bound to even attempt to be consistent in its opinions from case to case, and if it's all over the place in its verdicts and might well be accepting or rejecting a verdicts for different reasons from the justices (maybe they think the legal arguments about privacy in the original Roe vs Wade were weak but also think women should have an inherent right to abortion that legislators haven't provided them with), it'ś difficult to see how lower courts can be bound by it.
There's also a big question of who gets to vote, particularly for the cases where a state law is argued to be in violation of the federal constitution, which tends to be the more controversial ones...