Muscle needs to be a conductor because that's how the 'tense' message is signalled—witness Luigi Galvani and his experiments on frogs' legs: apply a current, watch the leg spasm.
An action potential (the phenomenon that underlies nerve impulses) is not just similar to an electric current; it is an electric current. From Wikipedia[1]: "An electric current is a flow of electric charge." It really is that simple. Any net movement of electric charge is a current. If I pick up a positively-charged baseball and throw it across the room (using insulating gloves so the charge stays with the baseball), that's an electric current. This isn't just a pedantic point, either. The moving baseball interacts with the Earth's magnetic field just like any other current, and an action potential also acts like a real electric current, because it is one.
However, you might contend that the electric current in an action potential moves across the membrane, not along it, and you would be correct. But that is because the voltages involved in the action potential are oriented across the membrane. The electrical conductors that carry this current are aqueous solutions of sodium, potassium, chloride, and other ions, and this solution will happily carry a current in any direction in which a voltage is applied. So when an external source (like an electric eel) supplies this voltage, the muscular tissue conducts the current. The ion channels in the membrane also ensure that the current is not stopped by the membrane as well.
Fat has high electrical resistance because lipids are predominantly hydrophobic and nonpolar, which means that they exclude both water and ions, both of which are necessary to carry a current in living systems. Muscle has a low resistance because part of its function is to carry an electric current.
Incidentally, if you think muscle doesn't conduct electricity, you might want to consider why the eel would even bother with all this electricity anyway. What's the point of zapping your prey if it doesn't even conduct electricity?
They are. That's the whole point of that experiment.
Moreover: cut the conductor and the muscle doesn't move. Which is why nerve damage is so crippling. You literally have lost the ability to signal the muscle.
You're describing a current without actually using the word current. Guess what happens when you have potential differences and a conductor. A current. It's literally the definition.
Smooth, skeletal and cardiac muscle cells and nerve cells do have some differences in their respective ion channel compositions. As a result, their action potentials will look somewhat different. And obviously muscle cells contract while nerves do not. Those differences aside, they are all are excitable tissues. Importantly, they all have voltage-gated ion channels. So they all will respond similarly when placed in electric fields.