The Hatch's spawn conditions have been confirmed in an official dev post. The entity renders visible to the client when either a minimum of two generators have reached 100% repair status, or the total active survivor count is reduced to exactly one.
The interaction state shifts from locked to unlocked autonomously only when exactly one survivor remains alive. The killer entity possesses absolute interaction priority over the hatch object, and a single interaction permanently seals the entity, forcing the exit gates into an instantly powered state.