I do like the other approach better. I think that "everyone's a captain" will make it harder for groups to get together and play, whereas if it was everyone working together on one ship, the game would encourage people playing together.
Plus, the best part: while the ship's crew could level up doing tasks (which I'm sure would be made fun through mini games or creative decision making), I would give the control for promotion (from Ensign all the way up to Captain) to the players.
The GMs could be Admirals or something.
At the same time however, this new idea sounds pretty good, with the create your own race, and it looks good. I also like that it is years after the TNG era (where I wanted what became Enterprise to be set) so that it doesn't mess with current series canon.
I will be angry if everyone ended up being Federation though. It would be good for PVP if you could choose to be a Romulan or a Cardassian or even a Borg, although if I were a betting man, I would say that all of the interplanetary factions in the game will be fighting against a totally NPC Borg fleet in most missions.