We might do well to think of the warp coils as the "propellers" of the ship, while the matter-antimatter annihilation chamber is the "boiler" and the plasma network is the "axle" (or, in certain types of seagoing vessel, the "electric network" or the "hydraulic network") that transmits the energy created by the boiler.
But the ship would also need a "turbine" to rotate the "axle" (or to run a "generator" for the "electric network" or a "compressor" for the "hydraulic network"). Does the m/am reactor pump out mere unadulterated obeys-known-laws-of-physics annihilation radiation and heat, which is carried in the plasma to the coils which then act as the "turbine" part that converts power to magical space warping? Or does the m/am reactor already pump out a little bit of magic?
There are pros and cons for both ideas. In the newer shows, especially VOY, the warp plasma pumped out by the reactor is a sought-for commodity. Why would all sorts of villains covet "axle grease"? The plasma might indeed already contain some of those "warp particles" that Janeway and Torres got all hot and bothered about in "Parallax", and thus already be partially magical itself. Just any plasma heated up by arbitrary means wouldn't suffice for activating the warp coils - only "warp plasma" would do the trick.
OTOH, we could use ENT as support for the idea that plasma is plasma and energy in plasma is just energy in plasma, and that the warp coils are our equivalent of "turbine" - the first part in the power chain where mundane energies are transformed into magical warp energies. When Archer steals the alien coil, we could argue this coil is a "keystone" piece of the entire coil set, the first coil through which the plasma runs, and the one that converts the incoming plasma energies (heat, EM, other real-world stuff) into magical warp energy. The first coil might not do much in terms of propulsion; the dozen coils in a row behind it would amplify the field into something useful, like additional loops in a coil magnet strengthen the field it generates. But the first, special coil would be absolutely needed in order to get things started.
Probably different starships, especially if built by different species, have subtly different setups. We could argue that the Fed (and Klingon?) system where antimatter annihilation gets routed through dilithium is one of those where the plasma is already magical and the coils are mere "propellers", not "turbines". But we know there are other kinds of systems out there. Perhaps Romulan singularity reactors produce real-world plasma, which gets converted into warp magic later in the process? Or perhaps the principles of the Fed and Romulan systems are the opposite of what's suggested above.
Timo Saloniemi