I see a lot of confusion about Yoyo boy. So let me clear things up.
If you've watched Goblin, you will know that there is a Godlike being that is one tier above Kimshin. This god has the ability to inhabit many different bodies, but in this case he chooses goblins nephew. The sole purpose of this god is to be able to the create rules that governs the ML, or in the case of KTEM, it's just a convenient way to explain away logic, because a god decided, a Deus Ex Machina as one poster has said.
In KTEM, we see a similar usage here, not surprising, since it's the same writer.
Yoyo boy is in essence, the flute. and the flute is a combination of the Fates (Greek Mythology)/and a god. I believe the writers intention of creating yoyo boy is the following:
The flute is an inanimate object, incapable of human behavior; but in reality the flute is sentient. The writer KES, wanted to humanize the flute, and the best way to do that is to create a sort of human avatar to display what the flute is thinking. So emotions like, the flute is calling out, or the flute is crying becomes easier to grasp when we consider the yoyo boy to be the flute.
When the flute is cut in half, so did yoyo boys powers and they show this weakened state in a few instances. Like when yoyo boy saves TE with a knife, and he shows his incapability to move thru dimensions. What limited powers yoyo boy had as a child we do not know.
We see the full extent of Yoyo boys power when the flute is made whole, and yoyo boy regains his true adult form. From this point on, we see that he gains godlike powers to enter into the subconciousness of different people, control memories of people, and change the time zone world own rules. We also get a superclip of yoyo boy imprinted into scenes with other characters.
These superclip of yoyo boy is intended to show us the audience that yoyo boy is actively changing fate. And this is where the contradiction may come, because there is a reason, it's called fate, it's suppose to be inescapable. Thus enters the godlike aspect of yoyo boy, where he even has the power to change the fate of people. So for example when we see SJ walking and sees TE in the window with her judo uniform, that is show telling us yoyo boy made SJ walk and notice TE, thus creating that particular fate where they meet.
Yoyo boy and the flute, also represent the Fates in greek mythology. There are 3 aspects of the Fates, namely the one who allots the lenght of the red string, the one who spins the string, and the one who cuts it. KES simplifies it by making it one being, yoyo boy. But I believe she may not have read the whole mythology, as the Fates typically deal with death, thus the cutting of the string, means you die, but this is not the case in KTEM. Yoyo boys even says "shall I break it", in the aspect of KTEM breaking the strings of Fate is interpreted as rewriting someone's fate.
And this is why we get so many happy endings in EP 16, because yoyo boy is able to change all these fates, that is the extent of his power. And yoyo boy, who is seens as god, also doles out god's punishment on the villains of the show by giving the villains poor fates.
And the flute, which is a representation of the powers of yoyo boy, and thus a god, is in the hands of LG.
You see, LG and TE were never fated to be with each other, it was only thru the use of the flute that they were able to meet in the first place.
But because LG now controls the flute, he is able to, for two days every week, change his and TE supposed fate. In many ways, LG and TE are cheating fate.
LG also becomes the yoyo boy, in the sense that he has the power in his hand to change people's fate. Just like he change Shinjae's fate by saving his life with the baseball.
And that is the ending we see.