... desires to die. Specifically at the hands of the demon, who he felt didnt finish the job after he was the dungeon master

There is a lot of foreshadowing up to this reveal such as:

-Mithrun's lack of ability to care about anything. His hygiene, his physical appearance, whether he is hungry or tired. All of this is common in those struggling with suicidal ideation and chronic depression. As Dungeon Meshi often toes the line between comedic and horror , usually comedic upon first inspection, horrifying with hind sight , i think this is best seen here:

...Why would he lie? Except subconsciously he wants to die. Whether through inaction , literally wasting away, or the dungeon monsters getting him. His preference would be for the demon to finish him off, but his actions sometimes show that the goal itself is death, no matter what methodology.

-Mithrun is covered in self harm scars. We know that he acquired them after his time as the dungeon master because of this story extra:

- this is further supplemented by his fighting style in the dungeon. Every action shows he is clearly a knife fighter, and yet he is never seen with a knife. He always improvizes weapons or is handed weapons by his comrades during combat.

Some examples of this:

This is even played as a joke in one chapter where Mithrun reaches towards the ground for a weapon before ultimately teleporting Kabru as his improvized weapon instead :

- which brings us to his teleportation abilities, it is extremely reckless for someone with no sense of direction and one eye leading to a lack of depth perception to rely on teleporting himself as his main trump card in fights. It shows a lack of value he puts on his own well being or survival

Although funny enough, the only time we ever see this go wrong in dungeon meshi is this extra where someone else, not Mithrun, is hurt:

- the reader is led to believe that Mithrun's goal is to defeat the demon for the majority of the story but when he does face Marcille and the demon , his eye glows its original gray, signifiying his desire and he says:

Mithrun is not addressing Marcille, he is addressing himself and ultimately he realizes his true desire is death just as he is about to recieve it

So YOU have chosen death. I'VE failed. The one who chose death has always been Mithrun , aware or not.

Marcille as the demon's proxy kills him. It is his first death in the dungeon, in any dungeon . Once he is revived , he is a blank slate with no desires left from his old self pre dungeon master. We are able to see what has always been hinted at, and Kabru confirms later on: Mithrun is able to gain new desires as long as he chooses to live:

Mithrun must live and seek new desires, he already has by surviving his death wish and then choosing to try and defeat the demon.(as shown by his eye glowing gray once more) He is capable of new desires and will live and find new meaning

After all one of dungeon meshi's major themes is the cycle of death and life :

The main cast learns to accept death, Mithrun to accept life.

Its why having a passively suicidal character realize their behavior and make steps to seek not death, not just surviving , but choosing to live is so impactful.

Once he accepts that, he is able to move on, his final words in the main story show that resolve. When talking about Falin he says this:

At the end of his quest for death, Mithrun chooses to live. As the text declares : your new life begins here