Well, stackexchange administrators believe that comments should not be used for anything remotely like an answer, I am moving my long pair of comments from the day before to an answer (and since the OP upvoted them).
The question really seems to be to be asking for epistemological arguments, yes?
Econometrics, I would argue, has been most successful in economic demography and economics of education, where adequately controlled data leads (all classes attended, job placement, sufficiently large samples to satisfy statistical assumptions, with detailed relevant background information, etc.) have led to meaningful conclusions. That is the place generally to look for successful econometric explanations.
In general, however, it's rather difficult to be impressed by claims along the lines that econometrics replaces coherent theory or is somehow more primary in explanation of facts. Consider the well know arguments (due to the W. Clifford and E. Mach) against a theory of history and history as a source of theory: --- no well-constructed experiments => no valid "law" inferences. (Typically, we simply don't have accurate preferences data to control historical data with, etc., etc.)
Also, without theory, demonstrated by other means, one cannot decide which measurements or facts are relevant for collection and testing of another theory. In the end, the reason a theory is confirmed by facts or discarded, depends partially on another theory. The experience would not speak for itself, although in some physics, where we can control for almost everything, it may. (Although physics is generally highly theoretical too.)
A shear stress test is the same shear stress test for everybody. But whether a farmer who contributed food to feed Wellington's army contributed to the outcome of the battle of Waterloo (this a factual question) would covary in answer with whether theoretically the food he contributed (if he did and what is a factual question too) is relevant to the economic dynamics. Otherwise, we get spurious and purely coincidental correlations that appear significant.
Also, there are no constants in economics --- because agents learn and preferences change. To say: A explains the dependent variable quantity B with extent C, if true, may be true by historical coincidence, without a strict reason. For, to say that it always has relation C to B, involves the assumption that people don't learn (which is false).
So I would argue epistemologically that more mathematics is required in economics, not less, which per se means less econometrics, if we aim to explain experience.
EDIT
The underlying evidence for the premises of theoretical arguments are (a) physiological-psychological results of well constructed experiment. (Extinction of the orienting reflex, say, corresponds to Gossen's Law of Satiety.) Or (b) tautological thought experiment. Not historical experience, the basis of econometrics, which is a higher-order analysis.
"A if and only if B" claims cannot be falsified by historical experience; that may only falsify "A does not exist" claims. Science, as C. Wolff is famous for defining, is the set of proven or provable claims (at least to the extent, as K. Popper argued, that all known alternatives are falsified). The purpose of science is explanation, for systematic and justified prediction (not merely coincidental prediction).
A classical analysis paper on what is required for knowledge (the ability to systematically predict, not merely guess): Gettier E, 1963, Is Justified True Belief Knowledge, Analysis 23(6),121-123. An open link. Medical guesses, as opposed to systematic physiological study, would be pragmatic belief, as opposed to science, in the language of Kant (Critique of Pure Reason, 2nd edition).
