OK - taking the questions one at a time. Full disclosure: I'm a member of the phonon tribe, but I'm trying not to let that cloud my response here.
"So, as far as I can tell, this is a simple, elegant, experimentally-proven theory explaining cuprate superconductivity. The theory and supporting experiments are at least five years old. But everyone still says that cuprate superconductivity is a mystery. What's the deal? Is there a problem or controversy in this theory?"
Sure: So why does [your field] need a whole journal, anyway?
Joking aside, yes, there are problems with this theory. From the experimental side, for the most recent evidence, the presence of orbital currents was excluded in the pseudo gap phase using NQR[dx.doi.org/ 10.1103/PhysRevLett.106.097003 ], and a muon spectroscopy study done on the very same sample that was used for the first polarized diffraction study shows that there is a substantial amount of magnetic impurity phase [dx.doi.org/10.1103/PhysRevLett.103.167002], making the clear cut interpretation of orbital currents ambiguous. The predicted fluctuations indicate one should see a strong quasi elastic response in bulk probes like neutron scattering upon entry into the pseudo gap phase, but as far as I know, there is no evidence to this point. This theory is seemingly absent an explanation for the observations from neutron scattering about spin resonances (gapped excitations that are observed at the AF ordering vector). It may be a red herring, but as the resonance is universal in unconventional SC and scales with Tc, some explanation is warranted. 70
Further, this theory is silent on the matter of electron doping, which also induces SC in some copper-oxide materials.
Does the theory explain only a small part of the mystery of superconductivity?
It provides correct order of magnitude estimates for the Tc and size of the superconducting gap. It explains some of the exotic behavior exhibited in the pseudogap phase.
Am I misunderstanding something?
Probably. There are hundreds (thousands?) of theories of HTSC, all of different merit, and all interpreting the evidence very carefully. As is pointed out in the comments, these things are pretty contentious, and I'll reiterate that the evidence has been refuted in several cases.
Now, none of this answers your primary question, which is whether or not this theory solves HTSC. The answer to that question is definitely maybe. It does get the order of Tc and the energy gap correct, and has supporting evidence. However, the data that these papers provide as evidence present some ambiguities in the interpretation of the experimental result in light of sample preparation issues and choice of probe. Fundamentally, a theory of high Tc must have some predictive power of the Tc, the energy gap, be supported by most of the experimental evidence, and to some extent, what the sample makers could put into a crucible and have pop out as HTSC. It is not clear that this theory has done the last of these two sufficiently, and some reconciliation needs to be done. 70