I always thought that when your mission begins it's basically top secret, just between you and Overseer. Nobody else knows about it for the time being, and one person being absent from the Vault for some time wouldn't attract as much attention as three or more people missing. More people missing, more questions raised, and eventually panic erupts when people realize that Vault is running out of water.
After a while, people of the Vault do find out about you, but by that time your mission is at its closure and Overseer had the time to spin the story in the right direction, so that they could kick you out.
On the other hand, Overseer might have actually sent a team after you have already left (but I don't so). Maybe he just didn't want to risk three lives instead of one for the time being. However, we find Ed's corpse just outside the Vault's door, so we're not the first one to leave. Remember, it's a single body. Talius was also sent by Jacoren before you. Maybe a policy was enforced where only person can leave at the time to ensure minimal loss of life?
Eh, regardless how you try to explain it, bottom line is that was a design decision from a pure gameplay perspective. Three characters basically exist to showcase three typical and balanced character builds and to give players who just want to jump in the game a two-click start. Never is it stated that Overseer Jacoren was making a choice between those three specific people. In the end, it's a single player game, so that was the main factor. Anything else is just headcanon.
Tim Cain kind of addressed the issue, but it's obvious the whole thing wasn't given too much thought. Basically, a single-player game, and that's it.