Is this possible? Many of my story ideas revolve around the main character being without his or her team.
Right now, no. We can only trick STO into spawning the player's Captain alone, or with a limited number of crew-members, but as the tutorial explains this highly depends on the map you're sending the player into.
It's a great trick and a very good tutorial about it. I do think however that eventually this trick may be disabled, especially if we're given direct control on the away-team size.
I think the best Cryptic could do for us, is to turn Spawn Points into something like what we see when we spawn enemies. A center location with 5 different mini-markers that we could set ourselves to dictate for each away-team member where he/she would appear on the map. This way we could simply edit those spawn-points individually and thus dictate how many away-team members the map allows and where they should appear when they beam in.