Yes, in time non-ADA compliant websites will be penalized in the search engines. It's already on their radar. But for now they are still negatively affected, just in an indirect way. Live traffic behavior (user experience) is now factored into Google's algorithm and is a ranking signal. Negative user interactions absolutely have negative affects on rankings.
The words in the tag should be more than a description. They should provide a text equivalent of the image. In other words, the tag should include the same meaningful information that other users obtain by looking at the image. In the example of the mayor’s picture, adding an “alt” tag with the words “Photograph of Mayor Jane Smith” provides a meaningful description.