There is a bit of a canon problem when it comes to the warp coils, as Enterprise had the coil as a small object that was easily transported from place to place, whereas when Voyager was shown to be doing maintenance on its warp coils, which were huge and located in the nacelles, they had to use shuttlecraft to move them,
link. Not to mention the episode of TNG that had a few scenes in one of the nacelles, where you could see all the warp coils.
From what I can tell, the M/AMR is where the power is generated, not where the warp field is generated. The core generates the power, which is passed to the nacelles via the warp plasma conduits, which then use the power to generate the warp field with the coils in the nacelles. It has been stated that when a nacelle is damaged that warp isn't possible until its repaired, if the core was generating the warp field then the ship should have been at least able to do warp 1, even without functioning nacelles.
It could be that the length of the core allows the matter/antimatter stream to be precisely tuned to generate as much power as possible, that way the Sovereign class would have a very efficient core, compared with the Defiant class, as the Defiant would need to basically throw as much M/AM at the core as it can to generate the huge amount of power it needs.
One thing I can't figure out is why a ship can't go to warp with one damaged nacelle, shouldn't the other be able to produce a field strong enough to move the ship, even if it wasn't as fast.