If I interpret your question correctly, you are basically asking what the practical difference is between interpreting a model as a reflection of reality and interpreting a model as merely a mathematical tool.

A mathematical model, at its core, is used to allow us to make testable predictions about our observations. Interpretations of that model into some kind of explanation about the fundamental nature of reality is more the realm of philosophy. That philosophy can loop back into producing more mathematical models, but the models themselves only describe behavior, not nature.

A model by nature is an analogy, and analogies are always reductionist. Like any analogy, if you poke it hard enough, it starts to fall apart. They make assumptions, they do their best to plug holes, they try to come as close as they can to mirroring the behavior of our observations, but they always fall short somewhere. Relativity and Quantum Chromodynamics are both good examples. Both are very, very good at describing behavior within certain boundaries, but fall completely apart when you step outside of them. (Both, to expand on the example, use constants that are impericaly determined, but we have no idea where they come from.)

The danger is in when you start to assume that a model of reality is reality itself, and you forget that it’s just a best guess of behaviors. Then you get statements like you first made. “Relativity assumes time is a dimension. The model for that works. Therefore time must be a dimension in reality. That must mean that not treating time as a dimension anywhere must be wrong.” That line of thinking, though, forgets that a model is only correct within the scope of the model itself. As soon as you introduce a new model, any assumptions made by other models are no longer relevant. That will pigeonhole your thinking and lead you to incorrect conclusions due to mixed analogies.