Those folk who actually have a backyard where they could bury their waste found that in areas of severe soil liquefaction that any hole you dug tended to fill at once with water. Also the yard may well be buried under a foot or two of contaminated silt.
As for building a composting toilet, note that:
a) If you didn't already possess the necessary materials, you had no chance to obtain them, as the majority of stores in the city were not open for several days following the quake. A great many of them will never be open again, at least in their original locations.
b) Once there were stores open, most of those able to get up and running quickly were on the west side of town. Access to and from the eastern suburbs ranges from difficult to downright impassable, and coupled with very heavy traffic, it's not unusual for journeys across town to take many hours.
c) Not wanting to stereotype these things, but the eastside is something of a lower socioeconomic zone. People there are less likely to possess personal transport. Limited bus services have only just gotten going again.

I've noted when using a bucket or hole in the ground myself that it may be okay for the young and able bodied, but your grandma is going to do far better with something she can sit on, and preferably not a portaloo 2 blocks down the street when she is in a hurry. Portable chemical toilets are now being distributed, which should help relieve the situation. Part of the delay in getting enough portaloos out seems to be due to the fact that they've had to import a pile of them - I guess there just weren't enough in the whole country.