What would it mean then for a Universe to not “run on math”? In this approach it means that in such a universe no subsystem can contain a model, no matter how coarse, of a larger system. In other words, such a universe is completely unpredictable from the inside. Such a universe cannot contain agents, intelligence or even the simplest life forms.
I think when we say that the universe “runs on math,” part of what we mean is that we can use simple mathematical laws to predict (in principle) all aspects of the universe. We suspect that there is a lossless compression algorithm, i.e., a theory of everything. This is a much stronger statement than just claiming that the universe contains some predictable regularities, and is part of what makes the Platonic ideas you are arguing against seem appealing.
We could imagine a universe in which physics found lots of approximate patterns that held most of the time and then got stuck, with no hint of any underlying order and simplicity. In such a universe we would probably not be so impressed with the idea of the universe “running on math” and these Platonic ideas might be less appealing.
Fair enough. I can see the appeal of your view if you don’t think there’s a theory of everything. But given the success of fundamental physics so far, I find it hard to believe that there isn’t such a theory!