It concur with most of what you say, but in some of your statements you persist in assigning primacy to theory. Saying, for example that "theory is central, because without it you have but a bunch of anecdotes" does not reflect some necessary aspects of science. One I already pointed out: when you apply the scientific method to a new field of endeavor of which little is known, no theory is available yet. By necessity you have to start with observations, then develop hypotheses and experiments to test those hypotheses. As these hypotheses grow in number and detail, theories with some predictive power may emerge.
Another way to see that experiments and observation should have precedence is this: nature does not care about theory, it is what it is. Aspects of what nature is can be observed through experiment and observation. Theories are crutches for human understanding: approximate models small enough to fit into our feeble minds. But theories are risky to rely on because even the most well-established theory may in the end be found to be flawed after performing an experiment in a domain where it has not been tested yet.
In the particular field under discussion, the actual situation is more complex because the experiments are interpreted in the context of stack of theories. To measure the degree of double-strand DNA breakage via a comet assay, as Lai and Singh have done, makes implicit use of a lot of physical theories and theories of biochemistry. This stack of theories is lacking a theory that provides a well-verified explanation of the precise biophysical and biochemical pathways via which electromagnetic fields can induce such excess DNA damage. There are multiple candidates for an explanation, and comet assay is exactly the kind of experimentation that can help you narrow down the candidates. As I mentioned, they found a reduced signal when the iron in the blood of the rats was first chelated. Precisely the kind of finding that can help you eliminate candidate hypotheses and move other hypothesis on towards the status of theory.
In short, they have been doing perfectly fine science. And even though a complete theory is not there yet, their experiments unambiguously indicate that electromagnetic fields can pose a mutagenic risk even when those EM fields induce no appreciable thermal heating and the photon energy (proportional to frequency) is in the non-ionizing regime: to establish that does not require a theory of the damage mechanism, instead it merely requires the theories lower in the stack that are sufficient to interpret the comet assay methodology. Hence, their experimental findings are very much worth knowing of and giving publicity to. Sadly, they have received repression and censorship instead.