Among the disciplines involved in science, Mathematics and Physics are the most epistemologically pure in my view. Thus I find your dismissal of math to be quite revealing:
In universality and information theory, it easily "trumps" the DNA evidence for a common ancestor.
Nor do I share your sense of what makes mathematical models useful. An example quoted by Cumrun Vafa was that Einstein was able to pull Reimannian geometry off-the-shelf to describe General Relativity. Which is to say, the geometry was discovered long before the warped structure of space/time was posited.
Likewise today, Max Tegmark's Level IV Parallel Universe Model is the only closed physical cosmological model known to me. It is a mathematical model - radically Platonist - that every thing in space/time is actually a mathematical structure that really exists outside of space and time.
Every other physical cosmological model (inflationary theory, big bang, ekpyrotic, cyclic, imaginary time, multi-verse, multi-world, hesitating, et al) - crash and burn on the issue of physical causality. In the absence of time, events cannot occur. In the absence of space, things cannot exist.
IOW, if you dismiss both God and mathematical structures as "real" - you are stuck with no explanation for physical causality. OTOH, if you dismiss God but accept mathematical structures as "real" - you are stuck with no explanation for the origin of the mathematical structures.
To recap some open questions for science:
That's funny. I talk to God all the time. He never mentioned you.
But you are not doing science, and you should stop pretending that you are.
Your discussion provides an interesting example of one of Burnham's laws, "Who says A, must say B." One of the most useful applications of this "law" is that it allows us to look at the practical implications of statements (A) which, in and of themselves, are not necessarily objectionable; and to see if the implications (B) are reasonable.
For example, one of the common arguments against ID theories is that they're "not scientific," apparently because it's supposedly impossible to tell the difference between naturalistic and "designed" effects. And yet, by that standard, the entire biotech industry is "not scientific," which is of course preposterous. This doesn't "prove" ID, of course, but it does neatly dispose of that popular argument against it.
You point out very nicely that there is an apparent necessity of some arguments to divorce mathematics from physical reality. I suspect the underlying motivation for this is largely emotional, rather than logical: for there to be a connection between mathematics and physical reality, is to suggest that there is a "reality" separate and above what we can observe. This is at least a necessary condition for admitting the existence of God. And that's a troublesome proposition for those who would argue that God (if there even is a God) plays no active role in the universe.
In essence, the debate is over whether mathematics is a process of invention, or one of discovery. The "dismissal of math" (the 'A' of Burnham's law) basically defines math as a series of "invented tools" that are developed to explain observed phenomena. And so by that standard, the 'B' must be that mathematics has no inherent physical meaning -- and thus, mathematical predictions of as-yet unobserved physical events are not meaningful. And it's stated baldly in the statement you quoted above: "Mathematical models and calculations are only useful if they accurately and correctly represent the data."
And yet this clearly fails to account for the fact that predictive mathematical theories are often developed well in advance of the observational evidence that confirms them (Einstein's gravitational theories being an obvious example). And think of the billions spent on particle accelerators which are often justified on the basis of searching for particles predicted by mathematical theories....
And, as you also point out, there are many examples of "interesting mathematical results" that have no immediate application, and yet just happen to drop nicely into place at some later point, in some complex physial theory.
The (B) statement that "mathematical models and calculations are only useful if they accurately and correctly represent the data," is clearly at odds with the practice of science. "The Data" are very often collected in response to the predictions, rather than the other way around. That particular (B) ends up being preposterous!