Yes, forces transmitted through fields act "at a distance", but is that really "spooky"? Do you think it is "spooky" that if you make a wave at one end of a pond, the wave reaches the other end? I don't. I consider the propagation of waves to be a "local" non-spooky phenomenon.

Disturbances in a field propagate through space similarly. A disturbance of the field at a point only affects the value of the field in the immediate spacetime surroundings, just like a water wave. I would call that "local" and non-spooky. Whether or not there is "mechanical contact", whatever that means, is irrelevant.

This is in contrast to Newton's theory of gravity, where the force of gravity was spookily felt instantaneously across space.