I think her storyline would be less appealing to a mainstream audience.
1. Dragons are the most prominent fantasy element in the show, although the white walkers are becoming more prominent.
2. Westeros and the people who inhabit it seem similar to medieval Europe and Europeans, which is more relatable to the audience. The dothraki are not relatable.
3. Most use of made up languages is in the Dany segments.
Point by point:
1. Yes dragons are fantasy, but they are easily one of the most popular parts of the fantasy genre. Many popular box office fantasy hits recently such as the Hobbit series and How to Train Your Dragon, even older hits like The Neverending Story and Pete's Dragon, focus on these creatures. When they are well done via CGI you add a Jurassic Parkish "wow" factor.
2. The Dothraki live in this barren desert on the fringe of any society. That whole storyline fits perfectly with the current boom of post-apocalyptic movies and TV shows. The rest of the people in the show have a "renaissance festival" feel to them that is certainly NOT mainstream. Hence all the people that love her, and by extension her story, more than anything else on the show.
3. I would argue that an English accent combined with a D&D dialect combined with loaded lines are much worse for the average viewer.
You see, with the foreign parts the average viewer will just think "it is foreign" and leave it at that. If they need the audience to understand either they switch over to English dialogue for the important parts (hence the graffiti in English) or they use easy to read subtitles. Extra bonus of foreign subtitled parts is that they can use REALLY simple to understand phrasing and words in these parts without dragging down the overall dialogue.
Meanwhile for everything NOT the Dany story, the regular person has to decode this odd dialect- without subtitles!- and figure out all the double entendres buried in the dialogue of almost every character.
Hence a huge chuck of GOT's popularity is built on people loving the show because of the blond female Abraham Lincoln with dragons, with other scenes that given them occasional sex, violence, and pretty vistas.