But don’t these mathematical models, however contrived, convoluted or compromised, yield results? Results that have applications in the material world, in the development of new technologies for example?
Almost without exception, the answer is “no.” For the bast 60+ years, basically the only application of the physics that is (or, before it emerged, was) the standard model of particle physics is to particle physics experiments and (by extension) cosmology (which incorporates the standard model of particle physics in its “standard model of cosmology”). This is not to say that there haven’t been plenty of developments along the way that have proved incredibly important, valuable, useful, etc., both within physics and elsewhere. Even when it comes to the sophisticated incorporation of what, historically, were two different methods for “taming” infinities (regularization and renormalization) into one renormalization scheme (in which regularization is a step in the process), the applications outside of particle physics have been enormous. Work that grew out of early QED, HEP, etc., led to the renormalization group (RG), and this along with other work was critical in our modern understanding of critical phenomena (pun intended) as well as with our conceptual understanding of certain types of complex systems. Symmetry breaking, path integrals, non-relativistic QFT, Wick’s theorem, Perturbation theory, etc., not to mention quantum field theory outside of its development and use within particle physics/HEP physics, have all contributed to out theoretical and/our experimental understanding of physical phenomena and some have had wider practical applications (condensed matter physics, for example, relies heavily on both QFT and RG, including the use of Feynman diagrams, virtual particles, etc., in the treatment of many-body problems).
But these “applications”, along with the mathematical and computational devices that are largely due (at least initially) to work in particle physics/HEP, are incidental (not to mention largely accidental or coincidental). In some cases it is not even clear exactly what was developed within particle physics first as opposed to borrowed from work elsewhere. And in all cases, the “applications” have nothing to do with the physical laws or dynamics of particles and the standard model. Not only that, a great deal of work was devoted to understanding why it is that we may never see any practical applications of quantum chromodynamics because we needed a theoretical explanation (color confinement) for why it appears that particles like protons have constituents, but we cannot detect these, much less manipulate them or use them for practical applications the way we do nuclear spin, positron emission, etc.
Many if not most theorists don’t think of most of the entities described by the standard model as more than formal terms in some Lagrangian required by the relevant symmetry groups and by gauge (field) theory constraints in general. As the general view nowadays is that the standard model is basically just another effective field theory, it is more or less a very, very, very expensive way of parameterizing ignorance IF we accept as true basic heuristic, hand-waving arguments used to justify would-be mathematical procedures applied to would-be mathematical objects.
But if we tell people this, then they won’t let governments give us billions and billions of dollars to build bigger and bigger machines that we tell the public allow us to probe the inner workings of the cosmos by revealing the secrets of the subatomic realm and so forth. Of course, for the theorist working in quantum foundations, the whole thing is a nightmare because the closest thing to an attempt to build a rigorous, coherent version of quantum field theory (algebraic QFT) has had no real success in particle physics, and all the issues we deal with in quantum foundations particle physicists have swept under various rugs in a myriad of ways using numerous tricks that even the founders (Feynman, Dirac, etc.) came to increasingly regard as problematic.
Has anyone tried doing this yet? Reimagined the physical world, in the hope perhaps of making a new kind of sense, from out of the impossible?
Yes. Its called quantum field theory. The reinterpretation involves demoting position from an operator and reinterpreting the operators from quantum mechanics to be particles that act on spacetime "points" we can detect via "events" in spacetime that correspond to e.g., cross-sectional calculations from colliders.