I'm guessing they didn't fully understand the responsibility attached to it. Seriously, if you offered an 8 year old girl the position of Queen, I'm pretty sure they'd jump at the chance- who could turn down fancy dresses and crowns and castles and banquets? 

Edmund for his part seemed pretty eager to be king when Jadis offered it to him. Even though it was mainly to get even with Peter, he later showed some excitement over the prospect of being in charge and creating "some decent roads." Maybe that kind of thing was still exciting to him even after he reformed.

The older kids probably accepted more for the reason you mentioned- they'd grown to love Narnia. Possibly they were also feeling more mature and confident about taking on the responsibility because of the whole Narnian air thing.  

Or possibly because... PLOT!  

I agree! But I also think about the fact that they earned the names Magnificent, Gentle, Just and Valiant over the course of ruling, and not at their coronations as the Walden movies show. 

Either is fine, but for this context I think that the fact that they did certain things that had them named their titles. Through their many years as rulers, they earned the titles and the thrones of Narnia, and came to appreciate the responsibility that came with each name and crown. In that way, they would come to accept the responsibility and not right away, which is reasonable.

Not to get off topic, but perhaps it was those specific things mentioned (stamping out the last of the Witch's army, saving the good trees, keeping fauns and satyrs from school and stopping busybodies and interferers) that earned each their name.

"And Peter became a tall and deep-chested man and a great warrior, and he was called King Peter the Magnificent. And Susan grew into a tall and gracious woman with black hair that fell almost to her feet and the kings of the countries beyond the sea began to send ambassadors asking for her hand in marriage. And she was called Susan the Gentle. Edmund was a graver and quieter man than Peter, and great in council and judgement. He was called King Edmund the Just. But as for Lucy, she was always gay and golden-haired, and all princes in those parts desired her to be their Queen, and her own people called her Queen Lucy the Valiant" - The Hunting of the White Stage, LWW

I'm just going to add a hot take here.

Even though they started out as foreigners in a foreign world (possibly universe), for whom the prophecy had no personal impact, it quickly became personal over the course of the story.  Tumnus could have taken the easy way out and Lucy would never have been seen again.  The Beavers mightn't have safely lead Peter/Susan/Lucy to Aslan at their own risk...Edmund mightn't have redeemed himself without the help of Aslan and the Narnians. 

They learned so much about themselves and each other throughout the process, that for reasons outside of logic, the prophecy had to be referring to them.  They needed to be Kings and Queens as much as the Narnians needed them. 

A political leader's rule is generally governed by customs, traditions, and precedents...inventions that come with age and life experience - neither of which make someone a trustworthy leader.  What Peter, Susan, Edmund + Lucy lacked in life experience, they more than made up for with all that Aslan equipped them to do.  If Aslan were real (and I for one personally believe he is, he just goes by a different name in our world), I think he could equip any kid to be an incredible leader and rule justly, valiantly, gently and ultimately with magnificence. 

This whole thing has got me thinking about King Josiah (one of Israel's most highly-regarded kings, whose reign began when he was 8...sound familiar? )

Posted by: @glenwit

They needed to be Kings and Queens as much as the Narnians needed them. 

Posted by: @glenwit

What Peter, Susan, Edmund + Lucy lacked in life experience, they more than made up for with all that Aslan equipped them to do. 

Yeah, I totally agree! Maybe the Pevensies were not ready before, but they were now, thanks to Aslan. I know it took years to become totally comfortable with the idea of ruling an Entire Kingdom.

@glenwit, you mentioned how the prophecy quickly became personal over the course of the story. You also mentioned how thanks to the other characters in the story, (Tumnus, The beavers) the Pevensies may have not ended up where they did at the end of LWW.  I wonder if maybe the Pevensies were comfortable with the throne because the prophecy predicted it.  Like, you mentioned, had the characters in the story not done their part, the Pevensies may have not gotten to be kings and queens. I'm thinking maybe it wasn't coincidental the Pevensies accepted the throne. I think that probably because they were always meant to be the rulers of Narnia, they naturally felt comfortable or inclined to accept or do it. 

P.s  (Yes, I'm familiar with The Boy-king Josiah. That story always amazed me!) 

I think the Pevensies accepted the throne because of how much they trusted Aslan. They knew he was good, and they saw all he did in defeating Jadis. Plus, they knew that the future of Narnia was dependent upon their decision.

I don’t think it was an easy choice for them. They left behind everything they knew back home in England, but I suppose they felt Narnia was their true home after all. Also, I don’t think they really even knew how to get back home. I guess they just trusted Aslan enough to know that they could trust his guidance, and took the throne with that in mind.