What advantages can a action-at-a-distance description have over the field view of forces, that makes (at least some) people try toward that description?
In classical electromagnetic theory one has both charged particles and the EM field. While state of a particle - its position and momentum - can be chosen freely, the state of the EM field is constrained a lot by the Maxwell equations, namely by the equations
\nabla \cdot \mathbf E = \rho,
\nabla \cdot \mathbf B = 0.
Thus the field is not an entirely independent object to describe; it is partially dependent on the particles and thus its numerical representation is partially superfluous in describing the state of the system.
On the other hand, even with these restrictions, the fields can be chosen in an infinite number of ways and it is not immediately clear which initial and boundary conditions should be supplemented to the equations to solve for, say, the motion of the electron around the proton.
If the particles are points, which is the simplest case and electrons may be such according to current knowledge, this formulation with one field has also very serious problem that it is not capable to determine force acting on the charged particle (the field is singular at the position of all point particles). This lead to great confusion about the equation of motion of a charged particle - see the most famous attempt, so-called Lorentz-Abraham-Dirac equation, which has serious problems though.
Action-at-a-distance theory developed in works by Tetrode, Fokker, Frenkel and by Feynman and Wheeler and others removes both the superfluousness and ambiguity of the field by replacing it by a new principle that determines how the particles move. For example, the above works concentrated on a theory in which the force acting on a particle is a sum of the Lorentz forces determined by individual half-advanced, half-retarded EM fields due to all other particles. More intuitively, one can set up such theory with purely retarded forces. As a result, one way or another, one obtains consistent equations of motion of charged point-like mutually-interacting particles. The mathematical function field is still useful as an instrument to find the force and derive the consequences of the theory, but it has no degree of freedom of its own, because it is entirely determined by the past motion (or future or both) of the charged particles. There is no "free field" in such theory; particles just interact with each other, which can be aesthetically pleasing to some.