This is the only way to get people to put their phones down
The only time I haven't used my smartphone for more than a few hours, was when I went to Cuba for a week and literally couldn't use it. This is indicative of most people's smart phone habits. Technology is so accessible, we need our devices physically removed from us to stop using them. After all, Internet addiction can present urges and symptoms in line with cocaine and alcohol dependence.
Short of going to a country without Internet, or just nailing your smartphone to the sidewalk, a reward-based system like the one implemented by Chick-Fil-A may be the only method that could actually curb cell phone use during meal times. Plus, once the phones are away, customers will probably realize how nice it is to live in the moment, and translate the behavior to dining rooms at home. Straight-up banning of phones in restaurants will never work, as there are always cheaters, and people don't like being told what to do. But giving customers the option, and a reward, is a brilliant compromise.