Ok, let me start by stating that I don't believe the combat in this game is fundamentally broken. I think it's ok...but that's the problem. The rest of the game is great, and combat is just ok. I keep seeing complaints about it, but many of the complaints, and their associated fixes actually throw major game elements out the window, or completely fail to consider balance.
A lot of the fixes I see make combat deadlier, and this is actually a bad thing. Deadlier combat makes hit and run strikes way more dangerous and thus tips the balance of the game towards the Empire. The rebels need the ability to strike, take out a target, and retreat. Deadlier combat takes that scenario out of play.
You can't tweak combat much without looking at the bigger picture.
Combat suffers from a few things. It can be slow (compared to the rest of the game). It's clunky (dice, cards, cards, damage, dice cards, cards, damage, remove pieces). And it seems to lack theme, which the rest of the game has in abundance.
Address the last one, and the other ones might be overlooked.
There is plenty of clunky elements of the game (that's why it takes 4 hours to play). Selecting your missions is clunky and time consuming. The probe deck/hand/tracking issue is clunky and time consuming. The multiple decks and hands of cards is clunky. But it's all ripe with theme, so you don't notice the downside.
Think about it. When you are playing missions, moving fleets, launching surprise attacks, you are seeing it through your mind's eye. It creates an interesting narrative to follow and retell later. Combat though seems cold and lifeless. Roll some dice, play some extra damage or block cards, move some plastic. You don't feel engaged unless there is a Death Star, some fighters, and some plans involved which add some real weight and theme to combat.
Honestly, the same can be said for when the Rebels are making those hit and run attacks to take out a Star Destroyer also. You imagine the raid in your head. The ships dropping out of hyperspace. The corvettes and Y's focusing on the star destroyer and the X's making a screen against the TIEs. The capital ship is destroyed and the commanding officer issues the tactical retreat.
Most combat is just plastic vs plastic which a bunch of red and black dice though.
What if everything about combat stayed the same, but the cards were actually interesting?
Swarm Tactics - If you have more fighters than the enemy, you get 1 extra black hit.
Noble Sacrifice - One of your destroyed fighters directed his damaged ship into the bridge of an enemy capital ship. +1 red damage.
Shield maneuver - One of your capital ships can be used to absorb up to 2 damage that was directed to another of your capital ships.
Redirected shields - The engineers on your ship were able to redirect the shields to maximize their effectiveness. Block 1 red damage.
Superior tactics - If your fleet command number is higher than the enemies, you can add 1 black or 1 red damage this round.
Defensive maneuvers - Your fighter pilots are well trained and difficult to target. Block 1 black damage.
Each card now has some theme to it, and they also have conditions, which adds even more theme. Now you play the Noble Sacrifice card and you recall the scene where the A-wing flies into the bridge of the Star Destroyer. You play Swarm Tactics and recall the scene with the dozens of TIEs attacking the Millennium Falcon.
Separate decks for the Imps and Rebels would help also. The Imp deck would have little fighter protection, but they would have things like swarm tactics. The rebels would have lots of blocks, and things that would help with tactical strikes. This would also add to the theme and flavor of the game.
Ground cards.
Tow cables - If you have 2 speeders, they can use their tow cables to cause 3 damage to a single AT-AT this round.
Anti-Infantry tactics - Imperial mechanized units are well trained for combat against Infantry units. +1 damage.
Anti-Vehicle mines - Defense only - Rebel units mined the best approaches to the base. 1 red damage to 2 targets.
Orbital Bombardment - Offense only - Capital ships in space make tactical strikes against defensive emplacements. +2 damage
Cannon fodder - Stormtroopers are often used as the first wave against anti-vehicle weapons in order to preserve the more expensive equipment - change 1 red damage to 1 black damage.
Again, the basic effects remain the same (a little extra damage or block a little damage), but the cards now add to the theme and depth of the game. This ties it closer to the rest of the game. And that weird silence after battle where cards are shuffled and everyone goes back to staring at the board is now replaced with hearty banter back and forth about how lucky their fighters were, or how the other player ruined your plans with their shield maneuver or how the tow cables completely tipped the scales of that battle.