Posting these scenes because I think it's useful to show what the Enterprise fighting a Bird-Of-Prey actually looks like:
View: https://www.youtube.com/watch?v=NhUEzTXvR-8
View: https://www.youtube.com/watch?v=VPz-6HuM8Sc
Notably, at no point are "firing arcs" even mentioned or relevant at all. In the first scene, the Enterprise cripples the Bird of Prey with two photons (which are homing weapons with a wide arc of fire) after managing to detect its energy transmissions in time. In the second scene, it has to endure several hits, before eventually managing to detect Chang's ship, and destroying him with a single torpedo. Overall the match-up is much more like a surface ship fighting a submarine than a dogfight. This is not a coincidence; the original script for Balance of Terror (where the Enterprise fights a Romulan Bird of Prey in the Neutral Zone) was IIRC adapted or heavily inspired by a made-for-TV-movie about a US destroyer fighting a German U-Boat.
Realistically, the biggest deciding factor for a Federation starship fighting a cloaked Bird of Prey is whether the starship detects the Bird of Prey before it has a chance to fire. Which is entirely to do with our sensors and computers*, and not really to do with our phasers. If you can find the Bird of Prey in the first place, you can just shoot a torpedo at it and blow it up, like depth-charging a submarine. Whilst Birds of Prey do not always attack using their cloak, it seems to be the preferred option when attacking large Federation starships.
This may be because a Bird of Prey or even multiple Birds of Prey trying to fight the a Constitution-class starship head-on would be disadvantageous. It is the equivalent of a U-Boat trying to fight a destroyer in a gun duel on the surface - a win condition for the destroyer. Also very relevant is this detail from the wiki summary of Balance of Terror, which I will repost and highlight since it contradicts some assumptions we have been making:
This might sound weird, but if you keep in mind that "Bird of Prey = Submarine" in terms of inspiration, it makes much more sense. U-Boats operating on battery power are much slower than surface warships. Similarly, a Bird of Prey which can only use impulse power whilst under cloak and has to watch its energy emissions is not going to be able to burn its engines white hot. Keep in mind that the Romulan Bird of Prey in Balance of Terror was also a few decades more advanced than the current state of the art!
So the entire scenario where a ninja swarm of Birds of Prey leap out of the shadows and then flash-step into the blindspots of our larger more ponderous ship does not really have any basis in how ships in Trek are actually seen fighting. There are scenes where small nimble ships evade fire, to be sure, but generally in a larger engagement, and I don't think "blindspot" has ever come up in any from the show that I can recall. There are other potentially more valid arguments for phaser coverage, but this specific reason which has been brought up a million times is basically like buying an amulet to protect ourselves from tigers.
*(Which unaccountably and completely unreasonably do not add to our Tactical rating even though they should, and have done in other Star Trek quests @Sayle has run on this very site!
)
*(This is also I believe where the whole concept of "impulse power" as a distinct thing from the warp drive which came from. Although other episodes sometimes refer to "auxiliary power" instead.)
View: https://www.youtube.com/watch?v=NhUEzTXvR-8
View: https://www.youtube.com/watch?v=VPz-6HuM8Sc
Notably, at no point are "firing arcs" even mentioned or relevant at all. In the first scene, the Enterprise cripples the Bird of Prey with two photons (which are homing weapons with a wide arc of fire) after managing to detect its energy transmissions in time. In the second scene, it has to endure several hits, before eventually managing to detect Chang's ship, and destroying him with a single torpedo. Overall the match-up is much more like a surface ship fighting a submarine than a dogfight. This is not a coincidence; the original script for Balance of Terror (where the Enterprise fights a Romulan Bird of Prey in the Neutral Zone) was IIRC adapted or heavily inspired by a made-for-TV-movie about a US destroyer fighting a German U-Boat.
Realistically, the biggest deciding factor for a Federation starship fighting a cloaked Bird of Prey is whether the starship detects the Bird of Prey before it has a chance to fire. Which is entirely to do with our sensors and computers*, and not really to do with our phasers. If you can find the Bird of Prey in the first place, you can just shoot a torpedo at it and blow it up, like depth-charging a submarine. Whilst Birds of Prey do not always attack using their cloak, it seems to be the preferred option when attacking large Federation starships.
This may be because a Bird of Prey or even multiple Birds of Prey trying to fight the a Constitution-class starship head-on would be disadvantageous. It is the equivalent of a U-Boat trying to fight a destroyer in a gun duel on the surface - a win condition for the destroyer. Also very relevant is this detail from the wiki summary of Balance of Terror, which I will repost and highlight since it contradicts some assumptions we have been making:
A cat-and-mouse game ensues. The Enterprise is faster and more maneuverable, while the Romulan ship has a cloaking device and immensely destructive plasma torpedoes. However, the range of these torpedoes is limited, and firing one requires so much power that the ship must decloak first.
This might sound weird, but if you keep in mind that "Bird of Prey = Submarine" in terms of inspiration, it makes much more sense. U-Boats operating on battery power are much slower than surface warships. Similarly, a Bird of Prey which can only use impulse power whilst under cloak and has to watch its energy emissions is not going to be able to burn its engines white hot. Keep in mind that the Romulan Bird of Prey in Balance of Terror was also a few decades more advanced than the current state of the art!
So the entire scenario where a ninja swarm of Birds of Prey leap out of the shadows and then flash-step into the blindspots of our larger more ponderous ship does not really have any basis in how ships in Trek are actually seen fighting. There are scenes where small nimble ships evade fire, to be sure, but generally in a larger engagement, and I don't think "blindspot" has ever come up in any from the show that I can recall. There are other potentially more valid arguments for phaser coverage, but this specific reason which has been brought up a million times is basically like buying an amulet to protect ourselves from tigers.
*(Which unaccountably and completely unreasonably do not add to our Tactical rating even though they should, and have done in other Star Trek quests @Sayle has run on this very site!
*(This is also I believe where the whole concept of "impulse power" as a distinct thing from the warp drive which came from. Although other episodes sometimes refer to "auxiliary power" instead.)