OTOH, I could see a kind of "warp beam" technology being signifigantly different from phasers, possibly certain types of exotic disruptor weapons whose effects may be both dramatic and devastating.
The Varon-T, for example.
You misunderstand. To explain what we see of phasers, in terms of FTL capability, they HAVE to go warp-speed, right? We're also explicitly told to be able to do that, they need to be surrounded by subspace, like radio signals for Subspace radio... hence, Phasers are /some/ sort of projection sent through a subspace bubble to hit its targets.
Yes, that's somewhat ruined in TNG and onward where ships fire from distances where you can literally spit at one another, but in TOS, it's really the only thing that makes sense.