Is the statistical interpretation of Quantum Mechanics dead?
The statistical interpretation of quantum mechanics is alive, healthy, and very robust against attacks.
The statistical interpretation is precisely that part of the foundations of quantum mechanics where all physicists agree. In the foundations, everything beyond that is controversial.
In particular, the Copenhagen interpretation implies the statistical interpretation, hence is fully compatible with it.
Whether a state can be assigned to an individual quantum system is still regarded as controversial, although nowadays people work routinely with single quantum systems. The statistical interpretation is silent about properties of single systems, one of the reasons why it can be the common denominator of all interpretations.
[Added May 2016:] Instead of interpreting expectations as a concept meaningful only for frequent repetition under similar conditions, my thermal interpretation of quantum mechanics interprets it for a single system in the following way, consistent with the practice of thermal statistical mechanics, with the Ehrenfest theorem in quantum mechanics, and with the obvious need to ascribe to particles created in the lab an approximate position even though it is not in a position eigenstate (which doesn't exist).
The basic thermal interpretation rule says:
Upon measuring a Hermitian operator $A$, the measured result will be approximately $\bar A=\langle A\rangle$ with an uncertainty at least of the order of $\sigma_A=\sqrt{\langle(A−\bar A)^2\rangle}$. If the measurement can be sufficiently often repeated (on an object with the same or sufficiently similar state) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.
Compared to the Born rule (which follows in special cases), this completely changes the ontology: The interpretation applies now to a single system, has a good classical limit for macroscopic observables, and obviates the quantum-classical Heisenberg cut. Thus the main problems in the interpretation of quantum mechanics are neatly resolved without the need to introduce a more fundamental classical description.
As Peter Shor correctly says, what is actually being measured in one experiment/measurement isn't the mean value – as the OP wrote – but one of the eigenvalues, and the probabilities of individual eigenvalues are given by the squared absolute values of the amplitudes. This has been true since the 1920s, Max Born got a well-deserved Nobel prize for the probabilistic interpretation, and there's no doubt it's the framework in which our world works. Otherwise the blockquoted text in the original question is right.
Quantum mechanics is epistemic because the wave functions etc. express observers' subjective knowledge, but in the quantum sense. It's also "as ontic as you can get" because there fundamentally don't exist any "objective variables" whose values would be agreed upon by everyone. Despite the fundamentally subjective nature of the wave function in quantum mechanics, the theory also guarantees the agreement between observers etc. whenever it is required by logic or experience.
All the recent and not-so-recent papers claiming that they have "ruled out" an ontic or epistemic picture have only ruled out naive classical hidden-variable models that have nothing to do with the reality, so none of the papers has any relevance for physics as the science that studies how Nature works and not how it doesn't work. Both ontic and epistemic hidden-variable models have been known to be inapplicable to the Universe for something like half a century.
Ontic and epistemic hidden-variable models differ by assuming that there exist some extra hidden variables behind the wave function etc. but they still assume that the world is fundamentally classical and that's the problem with both of these classes. To describe Nature, one has to use the correct quantum mechanical framework which is neither "classical and ontic" nor "classical and epistemic" because it is not classical! Whether quantum mechanics itself is labeled "epistemic" is a matter of terminology and I would say Yes.
Quantum mechanics is a deterministic theory. Lagrange's thought experiment still holds within it's framework: should you be able to specify the exact state of the universe, you would know precisely the state for all future time (to the extent non-relativist QM is right.)
The difficulty in interpretation arises in these processes of "measurement" and "preparation" where 2 difficult (and related) things happen.
The first is that we'd like to describe the system isolated from the rest of the universe and there's no guarantee that the state space of the system and the state space of the universe decompose into a direct sum nicely. They certainly do not during preparation and measurement.
Second, the macroscopic measurement system is a statistically described thermodynamic system. We know very little about it's state except the extensive and average quantities (Energy, Entropy etc.) that define it's thermodynamic quantities. So when particles couple to this complicated thermodynamic system, (during both measurement and preparation) the combined wave-function can only be understood in a statistical way. The combined apparatus/particle state is an element from the direct product of the two state spaces, and moves chaotically (ergodically) through this state space.
Measuring systems are peculiar and interesting because they have Hamiltonians that (to the extent that they are good measurement apparatuses) decompose into nearly a direct sum of the particle's eigenstates. The combined state vector moves from the thin bridge of state space between eigenstates into the vastly bigger state space dominated by a single eigenstate. When we project this state out into a direct sum of the the macroscopic and microscopic system, we find (unsurprisingly) that the wavefunction has "collapsed" into an eigenstate of that particular measurement operator. It's statistical, but deterministic.
At least that's how it seems to me. If I'm completely off base here, somebody correct me please.