I’m an Agnostic, but I’ll weigh in because I used to be a Catholic and was very interested in this subject.
You can imagine three courts. All interpret the law. Let’s call it Law X. Courts A and B have
binding authority - that is, the power to
impose an obligation of obedience - whereas Court C can only give an expert opinion with no obligation of obedience attached.
If Court A and B come to a conclusion as to the meaning and application of Law X, a person is obliged to obey their conclusion. To behave otherwise would mean that these two courts are no different to Court C, which lacks the kind of authority A and B have and thus its conclusions are not binding.
A person will only ‘obey’ Court C when they agree with the Court’s reasoning. In that sense it has a kind of “epistemic authority,” like a well known scholar, but not the power to impose an
obligation, like the other courts.
Let’s say that Court A always interprets the law correctly. Court B is like a fallible human court, one which will sometimes get things wrong, but its decisions are still binding. Most people I suppose accept that something akin to Court B is the best humans can hope for and we can at most minimise the chance that the court will get it wrong by following “due process,” but knowing full well that, given its fallibility, it will by definition get things wrong at times (even an the highest appellate level.)
Court A is what some Catholics think of their church as. Of course, there are two things that the Catholic first has to have concluded: they have to have correctly identified “Court A” and then when “Court A” makes a “ruling,” they then must interpret that ruling. But what they know is that
if they do correctly identify the court, and
if they understand its ruling, they will never be
obliged to follow a ruling that is incorrect. Court A will never make an incorrect, but still binding, ruling, whereas Court B is guaranteed to do so by its very nature - if it never gets things wrong, it would be infallible. That’s the unfortunate reality of our secular courts - we know that innocent people will ocassionally be convicted, but be do our best to minimise the rate at which that occurs.
I’ve heard some Protestants talk about their church being binding but fallible. In that sense, their church is like Court B in my analogy. But what are the consequences of that? They, like the Catholic, must still correctly identify the “court” and interpret and understand the court’s “rulings,” but they will be
bound, occasionally, to follow rulings that are objectively wrong. In a worse case scenario, a secular court will cause the execution of an innocent person, whereas a church with “authority” similar to Court B will oblige members to follow a ruling that places their eternal salvation at stake. If they claim that they will always “double check” or “verify” or “validate” the decision of Court B, they are in fact talking about a Court C situation - they only “obey” what the court orders what they agree with its conclusions, and thus the court has no real
binding authority.
That’s my (imperfect) analogy, and I am willing to admit I may have totally misunderstood the arguments of both sides. Each scenario has problems, because ultimately, we all must interpret and apply the rulings ourselves - as opposed to, say, secular courts where the executive ultimately enforces the decision and it is out of our hands. And then, there’s also the issue of what the law is - in my analogy, Law X would be the bible, but the bible is a collection of texts and therefore there is the whole separate issue of discovering what “Law X” is, or indeed, if Law X exists at all.
And binding authorities - of the religious and secular kind - all involve an element of circularity in their reasoning. If Constitution Y contains a section that says Court Z has exclusive authority to interpret Constitution Y, how can an entity that is claimed to be and viewed as Court Z ever authoritatively, and thus bindingly, interpret the section that gives it that authority? If you ask candidate Court Z “are you the Court Z referred to in Constitution Y?” their decision is only binding if they are in fact Court Z - if they are not, then they have no authority to make such a decision. They are essentially saying “because we say we are Court Z, we have the power to interpret Constitution Y, which says that Court Z has the power to interpret Constitution Y, and we conclude that when Constitution Y refers to Court Z, it refers to us.” They presuppose the very thing in dispute.
Excuse any spelling or grammatical errors, I’m half asleep at the moment and can’t be bothered revising my post. I am Agnostic after all

and don’t have a strong view one way or the other - not strong enough to get into a bit debate over it.