Dramasailor says:
So at a first glance (and for a rough summary here to facilitate brainstorming), combat works like this (correct me if I'm wrong):
Step 1: Engineers boost or repair. All starships do this at the same time, no initiative-type order present.
Step 2: Pilots make their movement decisions. For nonstandard movement (tricks, redlining, whatever) additional checks are made. Also, Science Officers scan/target at the same time.
Step 3: Everyone shoots, in the same order as piloting, although everyone gets to shoot.
Step 4: Resolve all damage.
As a means of speeding things up, I almost see it like doing a DM-rolled init.
Combat is encountered and round/turn based timing starts. Have the players indicate their overall goal (kill ship A, disable ship A, run away from the fight, whatever).
DM rolls piloting checks for all parties (players included).
DM posts call to combat and the piloting order.
Either Engineers make a call and post now, or they set a standard action of "I repair damage if I have it, otherwise I boost us". Then the DM can adjudicate which happens based on that.
During the adjudication of Engineering, if the NPC ships go first, their movement is addressed in the same post. Otherwise, the PCs address their movement.
Here's where I see it getting kind of funky, and that's that the players can post indicating their goal for this gunnery session whenever their teams piloting time comes up (I want to shoot A unless they turn tail, then I'll forgo shooting) or something like that. In an absence of that, the DM can make the decision as to what the ship does in order to advance their earlier overall goal. If they set out to obliterate the ship, they will fire on A every turn unless they intervene.
I think this may be too much on the DM, but so long as the players set up their goals at the outset accordingly, it may be OK. I still think this is a minimum 2 day/round kind of combat situation.
Now, if we step outside their general standpoint and way of thinking a bit, try this on:
DM rolls all starship Responsiveness (using Piloting as the baseline), and posts the combat order immediately. When a ship is up in the order, they take all their actions immediately. They engineer, move, and make gunnery decisions right away. Then, once all inputs are in, the DM resolves everything similar to the book (all engineering is done first across the board, responsiveness dictates who moves where when and what damage occurs when). Yes, it takes out a BIT of the tactical back and forth that could be there in a tabletop setting where rapid fire back and forth can occur in minutes rather than days, but that's ALWAYS been the turned-based TTRPG PBP trade off. Players and DMs have to be willing to do a bit of fuzzy fudging to make the story play nice.
You basically nailed the RAW procedure on the head.
I really like your final idea and I think I'll present it like this:
1. I would have players declare a default combat philosophy for their role
2. I would roll piloting/initiative
3. Everyone declares their action. If someone fails to respond withing a certain timeframe, I would take their round for them, trying to go along with their normal philosophy
4. I resolve all actions and do it all again.