Funny enough, in the UK/islands I've found it to be much more common for the 'door close' button to work.
As the other replies have said, the button isn't for you, but it does have the psychological side effect of giving you a feeling of control over your experience, which can help quell anxiety. This is also the reason mirrors are common in elevators; the space feels much bigger and less claustrophobic, and you can waste a few seconds while you check yourself out in the mirror which makes the trip feel faster.
However, the button will usually work in a hospital, as they want the door to be open for a long enough period that people who are less mobile can enter before the door closes, but at the same time there are often urgent situations where seconds count, so you need to be able to get the door closing faster in an emergency.
As the other replies have said, the button isn't for you, but it does have the psychological side effect of giving you a feeling of control over your experience, which can help quell anxiety. This is also the reason mirrors are common in elevators; the space feels much bigger and less claustrophobic, and you can waste a few seconds while you check yourself out in the mirror which makes the trip feel faster.
However, the button will usually work in a hospital, as they want the door to be open for a long enough period that people who are less mobile can enter before the door closes, but at the same time there are often urgent situations where seconds count, so you need to be able to get the door closing faster in an emergency.