I thought the junk collecting mission was kind of interesting, and encouraged using the piloting minigame, but the other two types of flying missions I would just grab whenever I got to a new system, and take the free money as a part of my delivery missions.
Overall, I feel like there's been several attempts at distractions from lulls in the action that I end up pushing aside or ignoring altogether. I find myself getting the crate stamping done right after bringing crates on board, even though it might be better stamina economy to try and squeeze them in in flight, since it's not likely I'll have time to stamp them all during travel time. I'm happy to take free money for being in a dust storm, but the last thing I want to do is postpone the next delivery or Onaris dropoff to pick up 1000ish credits that I'll probably get anyway. And whatever maintenance needs to get done tends to be put off until I don't have a timer to move cargo, at which point I do all of it at once before focusing on the next thing. I dreaded getting a hydroponics bay, since I knew that it would be another thing to juggle while missions were going on.
In my experience, the rhythm of the game centers around the warp drive. As long as there's gas in the tank, you can and should always be flying somewhere, and any component fixing you have to do should be manageable while still keeping stamina near maximum. Once you reach the end of a freight trip or Onaris "survey" hot-and-cold game, you'll be near zero stamina, but if you've still got gas you can keep going and trust you'll rest up by the next planet. Once you need to refuel though, it's time to refuse the next mission, and take some "shore leave".
You can't order warp fuel online, which means you don't want to do it while there's a delivery timer going, and to me that completely changes gears. Now I want to order more fuel, pack it on, and now I've got plenty of stamina left to go over my ship and fix everything back up. This low pressure time tends to be when I go after upgrade units. If there were NPCs to talk to and story to advance, I imagine this is when I'd do it.
You've talked about players not having anything to do while enroute, but I feel like in the frantic full-gas-tank-times these lulls last a minute or two tops, and the longest gaps tend to come from low stamina more than nothing to do. Have you considered adding sleep back in to skip dull moments and low stamina? Maybe this would be a good time for NPCs to come bother you with dialog? What if the player took the tutorial computer's advice, and just zoned out for a moment while staring at some cool space sprite for a bit?