Diaspora has an excellent space combat system that is non-trivial, but is still relatively fast to play. Ships are all represented as characters with several skills and stress tracks.
Ship crews are important because use of ship skills is limited or amplified by the associated crew skill. If a ship has a Beam skill of 4, but it's gunner has only a Gunnery skill of 2 the roll is limited to a +2. On the other hand crew skills can amplify a ship skill, which means if the crew skill is higher the roll is at ship's skill +1. This emphasizes both the ship and the crew skills, putting a little more emphasis on the crew.
Diaspora has an explicit Torpedo phase as one of the two weapons phases. Torpedoes are a critical part of Diaspora combat as they are the only completely offensive weapon, and don't have some of the downsides of Beams.
However Diaspora doesn't differentiate much between classes of ships as you would like, relying primarily on aspects primarily. It does have optional rules for fighters, but these are somewhat limited. You could augment this by using the scale rules from Fate System Toolkit.
Diaspora space combat rule'srules' real strength is their one dimensional representation of the relative positions and velocities of ships. This greatly simplifies the three dimensional space and widely differing velocity (in magnitude and direction) of spacecraft while still allowing for dynamic space drama and leverages the differing skills of pilots and navigators and accelerations of ships.
One thing to keep in mind when using Diaspora's rules for your purposes is that Diaspora is based in a fairly hard science fiction setting, where ships use reaction drives for propulsion and that influences the rules in some ways. You may want to adjust those rules if you're playing in a setting like Star Wars where reaction mass rarely comes up.
Read more at the Diaspora SRD
I have used Diaspora to run a game with non-trivial space combat, but not with the extendextent of space combat the question describes. I've have also used the space combat as a mini-game, and it was somewhat satisfying in that regard.