Yes, you can say that a theory that describes the world by differential equations, is also a local variable theory.
But in the classical physics, i.e. not quantum, the local variables are not necessarily hidden. The Newtonian mechanics describes the world using the observable variables masses, velocities, positions, accelerations, and/or angles, angular velocities, etc. No need of hidden variables. Electromagnetism also uses electric and magnetic fields, charges, etc.
In statistical mechanics we have some difficulty: we use positions and velocities of particles in a gas. These are not hidden variables, but we cannot keep track of the individual movement of each molecule and molecule of gas.
In QM (quantum mechanics) the situation is more difficult. There are observables, as $x$ and $p_x$, that, according to Heisenberg's uncertainty principle cannot be measured together. We call them incompatible observables. So, people thought that there may exist hidden variables that produce well definite values for both $x$ and $p_x$, even if we cannot measure $x$ and $p_x$ together.
Bell's theorem, more exactly the violation of Bell's inequalities, shows that we have problems with these hidden variables. The problem, however, appears when we perform measurements on entangled particles. Moreover, the problems appear when the particles are sufficiently distant from one another so as to be impossible for them to interact.
Well, Newtonian mechanics doesn't deal with entanglements. Thus, no worry for it. Maxwell's equations don't comprise entanglements. Again, no worry.
As long as we don't have to do with entanglements, it's O.K., the theory may be local. But, as soon as we try to describe entanglements in terms of local hidden, or non-hidden variables, we can have troubles.