Hearing your setup makes me wish I brought a ship with a larger crew! Cargo ship seems like it was not the right choice for this challenge, lol. I have my entire crew Researching.
As for what I'm doing, it's pretty simple. I have two teams of 3 (since the Cargo ship only has 6 crew). One team is researching the planet and the other team is researching the ship damage. None of my crew are particularly good at doing this, except for my Elite Syndicate Hacker who is good at everything.
I think that there should still be some sort of weapon(s) included in starters, even if they are weak. Everything is variable though and would take balancing probably. It sucks to get crew randomly generated in a hostile zone and you are unable to rescue them because a single enemy spawned early.