If you think there is confusion now about the switch to digital TV, just wait until the original deadline next month. The bill to delay the deadline until June is racing through Congress. A version passed by the Senate explicitly allows stations to turn off their analog signal anyway.
Many will take the offer. They've got contracts in place to move towers or make other changes. They all can save money by cutting the analog signal.
Grandma will see some stations disappear from her analog set. Others will remain. She'll be banging on the old tube, wondering what's wrong with it.