Dan Patterson mentioned this before:
I just picture the destinations being an intersection between the polygon(s) and the lines (roads) rather than the location within the polygon. Have you considered just producing an intersection as described or are you concerned that each intersection point might be associated with two polygons?
This would generate many points, one for each intersection of a road (over which one will enter the area) with the same value M. Doing a closest facility will return the drive time, drive distance for the nearest location. It seems that this "hack" could be a solution to get what you are after.
So, I would go with Dan's solution, although I still don't understand what this value M represents and why it is able to represent such large areas...