Didn't the Enclave try and get into Vault 101 and the Overseer disallowed them access? This could be a reason he doesn't want to open the vault; he fears the Enclave and, more importantly, their intentions with the Vault's inhabitants.
The security report detailing the Enclave's attempts to get in doesn't appear on the Overseer's terminal until after you return for "Trouble on the Homefront", so that doesn't seem likely.
I'm also inclined to buy into the "disaster involving Amata's mother" theory, and that's how I always imagined it. She might even be the one who owned the Armored Vault 101 suit in Moira's shop --- Moira described her as knowing nothing about the wastes, clearly just out of the vault, and that she never returned for her suit, implying that she met some kind of disaster.
That all happened 10-12 years before you leave, which means that Almodovar would've developed his "No Leaving the Vault" mentality a few years before your 10th birthday (where he is first featured in-game, and where he clearly exhibits his opressing nature). By that time, the decision to pretend as if the Vault was never opened is in effect, as evidenced by Palmer's birthday dialogue. There's no mentioned of what the Overseer's stance was on this prior to your 10th birthday, so there's nothing in the game that says that Amata's mother wasn't the reason for the Overseer's stay-in-the-vault decision.
The only other vault dweller mentioned on the outside outside is by Colin Moriarty, who was able to describe the "brainwashing" that went on. This particular Vault Dweller was mentioned as escaping "about 5 years ago", well within that 10-12 year "turning point", so it doesn't disprove the theory.
Oh, and more on topic, to support the "Vault needing a doctor" explanation, here are lines from the transition script, the lines that play during the "one year later" part;
JAMES: Yes, yes. Cardiology, Pediatrics, Psychiatry. Whatever your needs. You'll find I can handle, well… I can handle just about anything.
OVERSEER: All right. Just remember one thing. We need a doctor, not a deadbeat. Fail to meet my expectations, and there will be repercussions.