Why does Loki have a broken Crown
The Loki in the main 616 Marvel comics is not at all the character he is in the MCU. Loki was a villain, The God of Lies, The God of Evil and that's all he was allowed to be. Some writers implied this was something beyond Loki's control but he was nonetheless the villain. Comic books; Loki died a few times and he always came back. He died again in the storyline Siege, and then he came back but it was different this time.
Thor loves his brother no matter how much of a cackling Saturday morning villain might have been. When he died Thor wish to have his little brother back, and he did. But instead of coming back to life again Loki was reborn as an innocent child with no memories of being a villain or the reason he grew to hate Thor. Kid Loki was a mischievous but well meaning kid but he still had to carry all the baggage of what his adult self did. That said, there was hope things could be better this time. It almost seemed like it was different this time. Loki got to fight for good, he joined the Young Avengers and got to become a hero. But it was all a trick.
The old Loki took over the young Loki's body and what was kid Loki died.... sort off. Actually what happened was the two halves merged and created a new identity. Teen Loki, who didn't want to be a villain. But then Thor finds out the truth.
Kid Loki was good, things were supposed to turn out right this time. Who or what Teen Loki is doesn't matter because he's not the brother Thor is supposed to have. When Thor finds out the truth he beats the crap out of Loki breaking his crown and punching out a tooth in the process. Stopping just short of killing him only because of a promise he made to Kid Loki.
Loki made a deal with the all-mothers of Asgard. For each good deed Loki did a crime would be erased. It worked Loki's dept was paid.
Then ten years past and Loki would become a villain once again.... And the All-Mothers knew. It was easier that way. For Asgard it's easier to follow fate and to have Loki be the villain.
Loki is the God of mischief and he has no choice but to become the God of Lies and the God of Lies is a villain.
But what is a lie anyway?
Loki has to accept his who is and who he will be. The crown he wears is his. Like Loki himself it's been through a lot of damage, reshaped by everything it's gone through. It's not the same as it was before.
....and isn't a lie just a story we tell?
Loki, God of Mischief and God of Stories.
Loki can't change fate but he can rewrite it and choose who he wants to be