There are stories both ways. Cinderella, no doubt, preferred the ball to the kitchen -- in all her various manifestation. The heroine of East of the Sun, West of the Moon no doubt did not prefer the strange castle where she did not know what got into bed with her each night, and still less the quest over land and sea to recover him, to her old home.
Much, no doubt, depends on what incites the story. "Terrible bad things have threaten -- or harmed, or overturned -- your safety" is needed to push people out of their comfortable homes. A discontented soul is much easier to get moving, though some inciting incident is needed to make it plausible that they left just then. (Lends itself more to a character-driven plot, though.)
Of course, both ways have their problems. Lord of the Rings was leaving the comfortable home, and it was slow getting off the ground. (Frodo's desire for adventures like Bilbo's did not get much play.) Plus, of course (though Lord of the Rings had less of this), you have passivity problems with the hero. Sitting around content with what you have may be admirable virtue, but it's very poor aesthetics. Your character needs to want something -- which does, I suspect, help explain why so many of them start with an attack and the character charging off for revenge.
Then if the hero's trying to get away -- there is first of all the question of whether the character has real reason to get away. There are Special Little Snowflakes who think it's an injustice to have to do their share of the farm-work; there are Poor Abused Children who are the butts of unmotivated malice; there are people who are leaving other characters in a lurch (which is why so many characters are on the cusp of adulthood, minimizing their chances of doing this).
But even if the reasons are justified, there is the little matter of he can't get what he wants, or the story ends. Even if he wants adventure -- once he gets it, he has to have another motive. To be sure, staying alive works, but it's hard to make your hero active if all you do is throw trouble at him, so he's scrambling to react and ends up as passive a hero as anyone else who's pushed around.
Then you have to raise the question of why the hero is continuing the quest after it's become clear that it's going to be even worse than back home -- it has to be, to raise the stakes -- and the hero doesn't even raise the question of why on earth aren't I just dumping this and going home and eating humble pie?
Part of bittercon