Why is the issue only with reading advertisements? According to Shulchan Aruch (סי’ שז סי”ב ואילך), it is forbidden to read any unholy content on Shabbos, as a gezeira lest one read שטרות on Shabbos.
It is even prohibited to read a menu that one prepared for the meal. It is forbidden to read letters of greetings. It is forbidden to read a caption on a picture. It is forbidden to read secular books of parables or war stories.
The Shulchan Aruch clarifies that this issur is BESIDES for the general issue of bittul Torah and reading secular matters.
The Shulchan Aruch Harav (as well as Mishna Berura, etc.) writes that reading יוסיפון and the like (Jewish history) is permissible on Shabbos since it contains inspiration in Yiddishkeit.
Accordingly, it would seem permissible to read the articles with Jewish content or Jewish news, since they contain some level of inspiration.
However, the articles about national politics or historical info about wars, etc. seems to be completely forbidden on Shabbos.
This issur would seem to include most Zman articles, as well as much of Ami magazine, etc.