We need to define "emergency." You seem to be thinking an emergency is when somebody gets locked into the vault and needs to get out. But in reality an emergency door is intended to be a second entrance into the vault from the outside, in the event the main door cannot be opened. A fault in the main door would be a great burden to a bank. If it's the safe deposit vault then customers cannot get to their boxes and that might be a really big deal to them. If it's the bank's money/securities vault, they cannot conduct business.

The emergency door allows another way in, and allows a vault technician to debug the main door from the inside (or at least, try to get it open for business, then fix it). These were much smaller as a way of conserving cost. And, in the vault, there is less space that needs to remain unblocked than if a second full-size door were installed.

To maintain overall security, the emergency door needs to be just as secure as the main door. Hence it is the same thickness, has the same number of time locks, and usually the same number of combination locks, though the diameter is just large enough to get an average person through it.

These are no longer used as far as I know.