Is it possible to have a satisfactory physics which is content to describe everything in terms of operational (functional) properties? Could it therefore be the case that ontology is a useless concept leading us down a philosophical rabbit hole to nowhere?
Edwin Jaynes argued that Niels Bohr's original version of the copenhagen interpretation was exactly that: a theory that only makes epistemological statements and refuses to say anything about an ontological world. However (according to Jaynes) most people didn't really understand what Bohr was saying, so what we normally think of as the Copenhagen interpretation is a misinterpretation of the interpretation, in which Bohr's epistemological statements become ontological facts about the world.
In any case, it is certainly possible to interpret the formalism of quantum mechanics that way. In fact I would say that it's the only reasonable way to interpret the mathematical formalism. Quantum mechanics is basically a formalism for predicting the outcomes of experiments. Everything you calculate is an operationally defined probability of the form "if you do this experiment, these are the outcomes that can occur". However, the mathematical formalism has nothing to say about any ontological process that would cause those particular outcomes to occur with those particular probabilities. This is exactly why there is so much room for "interpretation" of the equations.
However, I would argue (along with Jaynes) that this doesn't mean we should throw ontological thinking out the window just yet. It might be that, for some reason, we live in a Universe that (in some yet-to-be-clearly-stated sense) fundamentally has no ontological reality. But then again, there might be an underlying ontological reality, and if there is then knowing about it would lead to new fundamental physics. We know from Bell's theorem and similar results that if this underlying reality does exist then it must have certain (arguably counterintuitive) properties, but its existence cannot be ruled out entirely at this point. Depending on your philosophical outlook you might consider the existence of such a thing to be more or less unlikely, but you'd have to be pretty hard-line to discount it entirely. If we don't look for it we'll never find it, so it seems wise to have at least a few smart people thinking about the possibility.
As an analogy, the Lorentz transforms were already worked out (by Lorentz) before Einstein, but their interpretation was unclear. We could just have stopped there - we had the equations after all, and so why not remain content to interpret them operationally? But Einstein's ontological interpretation led not only to a more elegant way of understanding the mathematics but also to matter-energy equivalence and ultimately to general relativity. It would be hard to argue in that case that the search for an ontological interpretation was a useless concept leading down a philosophical rabbit hole to nowhere.
It is my hope (not belief, just hope) that one day something similar will be done for quantum mechanics, enabling us to see, if not the underlying ontological reality then at least one level further down. This requires more than just an interpretation of the equations - it requires a new theory that makes testable predictions, while also reducing to quantum mechanics in some appropriate limiting case. It is my suspicion that many of the biggest unsolved issues in physics (and in particular the unification of quantum mechanics with general relativity) will not be fully resolved unless we can accomplish this.
Physics starts by questioning "why?" . Why does the sun set? Why does winter come? Why does fire burn?
It proceeds by formulating mathematical theories that not only fit the observations but also predict new ones. Then it proceeds answering "why?" : given the axioms and postulates it shows "how" the observations "why" wants described are arrived at. The question now becomes "why those axioms and postulates", at a meta level.
Suppose a string theory model fits all the known data and predicts new phenomena, a Theory of Everything, from which one can calculate how given its axioms and postulates all observations are correctly described and predicted, we shall reach the ultimate metalevel "why this TOE" which will no longer be a physics question.
All this can be encapsulated in the statement: ultimately physics answers by 'how' , 'why' questions are philosophical.
While ever finding, knowing or understanding the ultimate ontology is perhaps impossible, striving to move one more turtle down (or reach one lower-rung on the ladder) is one of the most fundamental motivating questions of any science. If it wasn't for the search for origins, we would be stuck at celestial spheres and prime movers.
What is missing in Physics is exactly Ontology. Physics in the modern form is about establishing a mathematical formalism about the natural material world that predicts and explains phenomena through determining quantitive relationships between things. But what these things are always eludes its grasp.
Feynman, makes this point in his popular book Character of Physical Law.
What is Being becomes a burden for philosophy.
It doesn't follow, from the fact that physics (allegedly) could describe everything in terms of properties, that there is no ontological work to do with regard to physics.
There is a massive recent (and ancient) literature on the ontology of properties: what are they, what are their existence-conditions, do they exist corresponding to every meaningful predicate, or just to those predicates that "carve nature at the joints". There are also substantial ontological debates concerning whether a world of properties alone, with no particular objects to instantiate them, is even coherent (cf. Russell's brief early flirtation with "bundle theory").
Quine, David Armstrong, and David Lewis ("New work for a theory of Universals") would be good starting points for more reading on this topic.
You ask whether it is possible to frame the laws of physics in terms of operational properties. But how do you know which quantities are operational? You could look for an explanation of what quantities are operational, in which case you are no longer solely using operational quantities since you are picking the operational quantities out of a set that are not all operational. If you don't look for such an explanation you may miss problems with your assumptions about what is operational.