99% of the discussions I've seen were purely on how to beat the quest. But not what the quest means or why it's worth playing. So, I'll start here.
With a critical eye, of course:
- Is Anju technically a pedophile? Or who cares because everyone's gonna die?
- Kafei was going to potentially leave Anju to die just to obtain a symbolic mask?
- Anju and Kafei let themselves die in Clock Town for pure plot convenience.
1) No, she isn't. Just like the others mentioned here, Kafei himself is an adult and was changed into a child by Skull Kid. So he's actually an adult that happens to be stuck in a child's body at the moment.
2) According to what I can remember Kafei's been missing for about a month before the events of the game takes place. After he was changed into a child, he went to see the Great Fairy to see if she could use her magic to help him, but instead he ended up becoming a victim of Sakon's thievery and lost his wedding mask. After this, he went into hiding and began attempting to find the man who stole his mask. He also knew that he couldn't officially marry Anju without his mask hence why he was searching for it. Not to mention they had promised to exchange those masks on the Carnival of Time when they were younger which only added to his desperation to find the mask in order to fulfill his promise. It was due to that promise that he didn't want to show up to empty-handed to see Anju.
After having a fruitless search for his mask, Kafei eventually had to face facts and decided to let Anju know that he was alright and that he was still thinking of her by writing her a letter. This in turn lead to the quest that Link goes through during the game.
So yes, potentially Kafei was going to leave Anju to die on her own just to get a mask that in retrospect didn't really mean anything in comparison to being with the person you love. So in that yes, he was being extremely stubborn about it and should've just let to mask go especially since the world was about to end. But it was his devotion to fulfilling the promise he made to Anju regarding the mask that made him do this.
3) Put it this way, both of them knew that they were going to die regardless of what they did or where they went. (Even Cremia knew this even though Anju and her family was seeking shelter there due to the moon falling.) Due to this they made the decision to die together. They both had did the things they were wanting to do; Kafei had fulfilled his promise to Anju to greet her with the Sun's Mask and the two exchanged their wedding vows on the Carnival of Time like they had planned and became husband and wife, so the only thing that they could do was be together in what would be their final moments. In the end, that's all that mattered.