On larger ships, such a system would be too powerful. The smaller ships in the Star Trek universe have the same problem smaller Starmade ships have. They just get creamed too easily. That's realistic, but annoying in a game. Cones of fire are too large already, making it impossible for small, fast ships to strafe out of a larger ship's line of fire.
That said, it sounds cool in a way, and I think it's actually possible right now. With the current wide angle of fire, you might be able to just put a non-moving turret flush with the hull of your ship. Unless your lighting is particularly bad, it should look fine. The downside is that you need more phaser banks than turrets to cover the same area. The advantage is that there's always something pointing in each direction.
I think the best solution is a configurable cone of fire from the muzzle of a weapon. Most servers will probably use a small one for balance reasons, while Trekkie servers would crank that thing wide open so AI-controlled phaser banks can fire 180 degrees or so.