The universal health care means that the extra insurance was not very expensive at all since it only covered the speedy access part. If you have money capitalism will always ensure you have speedy access to healthcare.
When a patient in the operating theatre at the boutique private hospital collapses or stops breathing or whatever, they don't have an in-house team to fix that, they'll deliver first aid (admittedly they'll be better at it than a lay person who took a two day course months ago) until the public system's ambulance takes the (likely unconscious) patient to a public hospital's emergency reception which can take proper care of them. Many of the doctors are the same people but even if the private hospital had the right people, and the right equipment it doesn't want the liability.