Yes, ultimately Egg will become king, but that's a long and winding road, and the subject for many a later story... which I hope to write after I finish all the millions and zillions of words I've still got to write for A SONG OF ICE AND FIRE.
The Targaryens have heavily interbred, like the Ptolemys of Egypt. As any horse or dog breeder can tell you, interbreeding accentuates both flaws and virtues, and pushes a lineage toward the extremes. Also, there's sometimes a fine line between madness and greatness. Daeron I, the boy king who led a war of conquest, and even the saintly Baelor I could also be considered "mad," if seen in a different light. ((And I must confess, I love grey characters, and those who can be interperted in many different ways. Both as a reader and a writer, I want complexity and subtlety in my fiction))
Lastly, some fans are reading too much into the scene in GAME OF THRONES where the dragons are born -- which is to say, it was never the case that all Targaryens are immune to all fire at all times.