Seems like the biggest problem is the lack of reserve capacity in phone agent staffing.

It probably would not be technically difficult for the airline to be able to mark specific itineraries as being affected by the cancelled flights, so that the web site could allow a one time free rebooking to the same destination with no penalty or fare change. But that would still not cover every situation (e.g. airline's own flights are sold out, so it needs to send passengers to other airlines). That would still run into the lack of reserve capacity in phone agent staffing.