As a reader, it's useful to see that an article is a featured or good article, as it helps indicate level of quality and peer review of the article.

As an editor, it is motivating to see the featured article star on something you worked on :) Or to see some given article is not, then maybe I want to work on the article to get it to good or featured article status.

Thus, the badges should carry over to mobile web and the app and be displayed somehow.

This should be very easy to do now since Wikidata stores badge information and we now add a page prop for pages with badges. (see T111025 where we are even making sure the page prop is populated)