Actually, I am an engineer, mathematician, and scientist (all three, depending on what hat I'm wearing). Don't confuse engineering and science, which merely USE mathematics with mathematics itself. A lot of the simplifications and empirical equations used in engineering and science are a way of dramatically reducing the computational complexity of a problem (which in many cases would otherwise be intractable) while still delivering adequate results. Mathematics is relatively clean and neat; the application of it in the real world is tempered by practical factors.
But mostly, it seems that your argument is with engineering and science. Math just describes relationships, and not liking the relationships it is used to describe isn't an example of inadequacy. It describes elegant relationships just as easily as it describes inelegant ones (a property which tends to indicate that it is quite adequate indeed).
A good test case for this statement (relating to the correspondence between mathematics and the universe) is the N-body problem. Current mathematics has no closed-form solution for it, despite which the N-bodies go blithely on their merry ways.
A basic philosophical question here is whether a true mathematical description of the system should be closed-form. One is tempted to say "yes," though the underlying requirements for the statement would require some very basic proofs which do not as yet exist.
Be that as it may, the present most-accurate mathematical approach is to integrate the equations of motion. Numerical integration is an explicit approximation of the problem to begin with. Beyond that, even an exact integration would be only as accurate as the force models (approximations), and also the truncation of forces being modeled.
Math just describes relationships, and not liking the relationships it is used to describe isn't an example of inadequacy.
This approach tends to treat math as something akin to a "force of nature," whose principles merely await discovery. In that vein, one can imagine a combination of perfect integrators, perfect models, and the inclusion of all perturbations, that would give exact results. However, these all presume perfect knowledge of the system -- not to mention assuming the availability of the mathematics necessary to implement the prediction. It seems quite unlikely that we can ever assemble such knowledge.
Bottom line: whether or not it's ultimately "perfect," mathematics is inadequate, because we are inadequate.