So a shower is science if the warm water inspires someone to come up with a new theory? That is a terrible place to draw the boundary; it opens up a can of semantic worms about what 'using' means. Major scientific results that are most easily expressed using calculus can be derived using processes other than calculus.
> The problem with the data first approach you are advocating for is that it almost never results in a good explanation
Science doesn't provide good explanations at the best of times. I'll willingly bet not 1 engineer in 100 can explain gravity. We just know if we measure mass we get predictive estimates of velocity. That isn't an explanation; that is reliable magic. In fact there have been a couple of centuries where even the scientists were basically just plotting data and curve fitting.
Math is a tool, yes, but it is a tool that is very useful to do science in and the act of creating and distributing a tool you use to do science is a large part of science. A shower on the other hand has no direct relationship to science, you don't need to be a physicist to make a shower, it is just a commodity. Calculus however needed to be discovered by a physicist as the math was too strange for mathematicians to find but simple enough that two different physicists found it in parallel.
The problem with the data first approach you are advocating for is that it almost never results in a good explanation, and almost all the value in science lies with the good explanations we have built up throughout history. Formulas are mostly useless, you need an intuitive explanation or people won't be able to build on top of it.