Elsa - “Protector of This Dominion”
One of the things that some readers fail to realize about Elsa – but this is a vitally important point – is that in her flight from Arendelle, she is, in fact, fullfiling the primary oath that she took when she was crowned monarch: and that is, to be the “protector of her dominion.”
According to the novelization – in the film, this speech is uttered in Old Norse, so for this we need to refer to the novelization – the bishop proclaims Elsa to be "The undoubted queen, protector of this dominion," as he bestows upon Elsa the crown jewels. And Elsa – whose sense of responsibility and self-sacrifice always supplants any concerns for her own well-being or happiness – takes these worlds very seriously and, by her actions, fulfills them.
Why does Elsa leave the kingdom?
Because she becomes a danger to it.
It is precisely because of the oath that she took – the oath to be the “protector of this dominion” – that she exiles herself.
She is literally defending the realm by removing herself from Arendelle, because she herself has become the biggest threat to Arendelle.
As a little girl, when her powers were minor, she nearly fatally injured her sister – and that happened at a time when she was only slightly disconcerted.
Ever since then, as the film clearly establishes, her powers have been growing. And she cannot control those powers even under the best of circumstances.
But now – suddenly – when she is exposed, she stands before her entire kingdom and presents a threat to every man, woman, and child in Arendelle.
In the glove scene, she witnesses her powers erupt as threatening spikes pointed towards everyone. If they had been just a little bigger, people could have been impaled.
Again, remember: she nearly killed her sister once already, and since then, her powers have only gotten worse. Now, suddenly, all of her subjects stand right before her; now no longer is it just her sister in danger, but everyone in Arendelle.
When she asks the Duke to stay away (for his own safety), ice shoots uncontrollably out of her hand, causing him to fall. The Duke and his men could have been injured right there. (It happened to her sister, after all.) The Duke is a panicky sort, but he is right is his assessment of the danger that Elsa poses.
Elsa, therefore, recognizes herself as a walking weapon of mass destruction and a threat to any geographical spot that she inhabits. (She obviously has no way of knowing or even guessing that her power extends far past where she is, to even to where she is not.)
For the good of her people, for their safety, then, Elsa removes herself from the kingdom.
In performing this self-sacrificial act, she is not abandoning her royal oath. Rather, she is fulfilling her oath – her oath to be the “protector of the dominion.”
She is protecting her dominion from its greatest, most immediate threat.
She is protecting it from herself.
She clearly has no other options. What could she do – stand around, attempting to get herself under control, and possibly blast people in the meantime, as she once blasted her sister?
The only fault that anyone could possibly lay at Elsa’s feet is that perhaps she should have exiled herself from Arendelle even sooner.
But that is an unreasonable expectation to have of a teen girl whose parents have just died. And besides, she clearly felt the responsibility of at least trying to fulfill her duty as queen without endangering anyone.
It was a reasonable attempt. It failed. Leaving her with no other option but exile.
The only other solution available to her apart from exile is suicide. But if Arendelle is a Christian land (as the religious trappings of the coronation ceremony indicate that it is), then suicide is a damnable offense and absolutely forbidden under any circumstances.
Therefore, exile is the only option left to Elsa, to keep her people safe, no matter that this presents her with the prospect of a lifetime of isolation and banishment. But because of her noble nature, because of her impulse always to put the safety of others above her own wants, she makes this terrible sacrifice and removes herself from Arendelle.
[More below the cut — warning: minor spoilers.]