You’re forgetting an extremely important detail here: the Bird-of-Prey exploited the Enterprise’s shield frequency modulation to do a lot more damage than they would have done normally. The Bird-of-Prey fired and did significant damage with the leading hit, while the Enterprise’s retaliation did nothing.

Also, you’re wrong about the Bird-of-Prey only having forward firing arcs. There are definitely aft torpedo launchers. If you recall (or rewatch) the torpedos were what were doing the most damage to the Enterprise.

As for the shields, you have to remember that the Enterprise D was never designed to be a warship, and thus does not have any armor to supplement the shields. Without the shields, the Bird-of-Prey would have most certainly won.

And although the Enterprise definitely outguns a Bird-of-Prey, they have tended to be rather tenacious ships who are often underestimated. For instance, at the outbreak of the Klingon Civil War, two Birds-of-Prey nearly defeated a Vor’cha cruiser, which should easily outgun two Birds-of-Prey. They were also very successful against the Jem’Hadar. During one such conflict, a Bird-of-Prey was ambushed by three Jem’Hadar craft and managed to escape in one piece. During that same battle, a single Bird-of-Prey overwhelmed the Cardassian outpost on Korma, disabling the planetary defenses and destroying every building.

In contrast to the Enterprise’s lack of hull shielding, the Bird-of-Prey has a rather tough hull for its size. Even the most vulnerable spot, the underside of the hull, was able to withstand sustained fire from a Cardassian freighter with the shields down, and a system-5 disruptor took two shots to breach the hull.

The reason you thought of it is because you’re not a seasoned Starfleet officer. Underestimating a Bird-of-Prey is a poor idea, especially when they initiate with such an advantage.