Well, I don't know for sure. That seems to be the way they do it in TNG -- we know the impulse drive is fusion-powered -- but I don't know what the intent was in TOS.
I mean, I don't think TMP was saying that phaser power was coming from the warp engines for the first time -- what they said was that the new design "increases phaser power by channelling it through the main engines." Channeling through is different from drawing from -- it implies that the mains are used as a booster rather than the original source. Or that the same power circuits are used for both, perhaps because they channel power more efficiently.
Now, the problematical thing is that the intermix chamber as shown in TMP has no visible connections to anything except the impulse engine and the warp nacelles -- plus whatever's at the bottom of the vertical shaft, which is evidently the matter and antimatter supplies. Other than that, it's just an unconnected tube. So where is this channeling of power taking place? I suppose it could be somewhere up in the saucer, since that's where the main phaser banks are, but what about the phasers on the engineering hull?
But the lack of connections does suggest that other ship systems -- like the deflector dish, say -- are powered independently.