I think you made reference to how mathematics by itself is not a minimal guide for physics. Many mathematical areas are vastly complex systems, which might not serve as an effective foundation to reality.
Maybe a part of the problem is that we do not have a "unified theory of mathematics," assuming such a thing is possible. I think the foundation of the universe involves zeta functions, 8-fold periodicity related to Bott periodicity, homotopy indices, Langland number theoretic correspondences and so forth. This may involve some unification of some subjects in mathematics. I am not sure if this is comprehensive though. Physics in one sense involves working on a similar type of problem on deeper levels, while mathematics often involves pursuing the study of entirely different sorts of structures. These new structures can come into play with physical problems, but the method of thought is often very different between how a mathematician works out the consistency of some type of structure, and how a physicist frames a type of theory or solves phenomenology.
I would agree that Tegmark's MUH does not appear to satisfy certain minimal conditions we would prefer. In that I would tend to agree with you. I am not sure if this is exactly a proof though. For all we know our requirement for simplicity could be a bias that is wrong. Maybe the universe is vastly complex at is foundation. There are areas of mathematics that have physical implications which involve a huge level of complexity. The universe might in fact have some extremal Kolmogoroff complexity condition, which means the foundations are not only bewilderingly complex, but unknowable. I am not saying I think this is the case, but on the other hand I do not know that this is not the case.
Cheers LC