That never made sense to me either. The best I could come up with was that the shuttle had a fuel synthesizer that couldn't be used until Scotty powered it with a phaser.
The dialogue above doesn't outright contradict that, but it only works if you assume Spock and Scotty are leaving out half the details.
I think that makes the most sense. When I watch the episode, Scotty appeared to be using a hand-held device to "adapt" the phaser. Then he held the phaser upside down and into floor bay aimed forward and continued to use the hand-held device which presumably was triggering the phaser to dump it's energy in a non-destructive ("adapted") setting into the reactor to generate more fuel.
Since "The Naked Time" shows that a phaser can emit energy without a visible beam, I think it could be very well that the phaser was firing or transferring energy through it's emitter.