Can't a driver just be added as a named person onto the vehicle's annual policy?
If the annual policy provider allows this then absolutely, however, any incident may affect the no claims discount of the annual policy holder. By having a separate temporary policy the vehicle owner will have peace of mind that any damage to their vehicle will not impact their annual policy.
As a driver I already have fully comprehensive insurance. Can't I just borrow someone's car without arranging additional cover?
Sometimes comprehensive cover will allow you to drive other cars. However, it may only be an option in particular circumstances, such as an emergency, and there could be certain restrictions. Also, it may only provide cover on a third party basis, so any damage to the vehicle you are driving would be your responsibility.