It seems like the biggest obstacle to multicrew gameplay is the current shield mechanic.
The one shield per core rule means that it's always better to have single pilot ships. There are 2 ways around that; remove the shield advantage for solo ships or add an advantage for mult-crew ships.
One way to remove the shield advantage is to limit shields by number of active turrets. Since each gunner module already requires a pilot you would then have the same number of shields no matter if you crammed all your pilots in one ship or spread them around in individual ships. This would also allow for some additional mechanics such as forcing a (pro-rated) shield restart when you switch turrets.
One way to add advantages to multi-player ships is to provide synergistic roles on a ship. You could have non-combat roles (eg mechanic, shield-ops, energy ops, etc) that give you bonuses but require a player dedicated to some piece of equipment. An other option is to have some sort of network effect for guns (eg if you have multiple guns they get bonuses like tracking, aim cone, the number of targets you can keep identified or the likelihood that multiple shots hit the same system)