From what I've read the difference between independent Iceland and Norwegian Iceland was that Norwegian Iceland said it was Christian and frequently told the Norwegian King he couldn't do stuff.
As for Sturlungaöld, that's not really comparable to leaving the island. The English Kings had a much more centralized state then any Icelandic ruler of this period, frequent wars that the English fought France, and strong feudal oaths guaranteeing them troops.
Yet the English nobility made a point of dragging it's heels on French campaigns, creatively re-interpreting it's rights so it could stay away. Going on campaign in a foreign country is a big deal.
In Iceland, where the typical response to somebody complaining about rape or murder was "you should kill that guy, the Althing will declare him an outlaw so he won't be able to legally kill you back" actually leaving the island meant leaving your entire family open to rape/murder.
OTOH staying on the island to fight the very people most likely to rape/murder your family because you're allied to the other clan...