You are correct, saliva does contain enzymes that start the breakdown of proteins in food. The process is sped up by chewing. Mucous does not have anything to do with digestion, it is merely a lubricant.

Greenies do break down in the digestive system, but like any other food have to be chewed and broken up for digestion to occur. There are many foods that are untouched by digestion if they are not chewed.

News reports dealing with consumer affairs issues have a tendency to be driven more by sensationalism and ratings than truth. For example, one day coffee is a major cancer killer, the next it's a breakthrough medical miracle. The same has happened with alcohol. Various makes of cars have been named "dangerous" by news reporting at different times. Childrens toys, household appliances, the list is endless. Accuracy doesn't count, sound science doesn't count, all that counts is the rating for the show.