Of course, Heisenberg's uncertainty principle is a mathematical description of observed behaviour so one could rightly argue that it doesn't really explain anything - it merely describes things.
> There are literally infinite ways to create equations that satisfy measurements.
There really aren't. You seem to be thinking of something like in the movie The Number 23. But we're talking about equations, not numbers. Take Newton's Second Law (f = m • a). What equation can you write that expresses that relationship that can't be simplified to f = m • a?
As for equations, notice that many equations in Physics have a constant which turns a proportion to an equation. This is where you have leeway in constructing more or less arbitrary equations based on the variables you think are important enough to observe.
Coming to your example, Newton's f = ma equation is really f/m proportional to a. The units are chosen carefully to make the constant 1. This works under the assumption that mass is constant and acceleration is measured measured from a non accelerating frame of reference with non relativistic speeds. So, yes there are several other ways to write that equation.
IIRC, in this context the mass m is considred as a proportionality coefficient. Such that the force is propotional to the acceleration.
Sure, rewriting this would fix the constant to 1, but this introduces a concept of specific force, force per unit-mass.
So, yes, it's possible there is some physics "beyond" the one we know, and yes in order to go beyond what we know we have to consider the possibility that some of the stuff that lay at the foundation of the current physics is wrong (or, correct up to an approximation). And many working physicists are well aware of that fact and they do consider all the options on the table.
The problem is, you have to find something to replace it and it has to work.
Thinking about all crazy things is great. "Temporarily" throwing away some assumptions can be a productive thinking tool. The Heisenberg principle (like many other things) can be both something you want to keep and use as a foundation for other explorations, and it can be something you question. The field is made up of many people, not everybody should be working on the same thing on the same assumption; I think "putting ideas at rest at this point" should be relegated to old theories that have been fully superseded, and even then they can be still useful: even Newtonian mechanics can still be quite useful even if we know it falls short.