It is said that if a couple united on the day of the festival and dedicated a mask as a sign of their union, it would bring luck. So Kafei and Anju are doing it for luck, not because they have to.
The "restored" Kafei
The last two lines in the article are needlessly misleading. They're probably trying to imply that you can see the "adult" Kafei, but after going through the effort to do that quest and kill Majora on the same set, the view of Anju approaching was still in first person from Kafei's view, and Tingle was still blocking the sight of them walking together.
I don't have any concrete proof to say that there isn't one, but the anecdotal evidence supports that no adult model for Kafei was made, and hence why they set up that portion of the ending clearly intends for us not to see it entirely.
I'm not going to edit this article myself, since I think a better thing to say in place of the very last line would be to describe what happens during the ending if Link fails to get this mask entirely, and since I have never played through without getting the Couple's Mask, and don't have the time or desire to play through just to see at the current moment, I'll let someone more knowledgeable make the appropriate changes. Hopefully this comment will help prevent any reversions back to saying that there is some way to see the "wedding" go entirely unobstructed in terms of our view.