But the point is, as I understand it, that the choice is between a value-laden state and one that is not value-laden (or is less so).
In this case there is a very good reason to choose, but there is no way to choose, because the two options are equivalent.

He chooses between drawing lots and dying.

And if not for the option of chance, then he really would die.
In other words, even a person has no real way to break the symmetry.

I really do not understand this bizarre discussion. Even if he were paralyzed and unable to move, he would die. So no, he will not die, because he will draw lots.

Rabbi Michi,
if those are the two options before him—chance or death—then how is this connected to the question of choice in a symmetrical situation? Those options are not symmetrical at all.

Of course I am not bothered by his fate—whether he dies or not—but by the principled question: given the assumption that a person has choice, can he choose just arbitrarily, or only between things that involve value?
The possibility of dodging the question by casting lots does not solve the question.
And even if he decided to cast lots, the questioner here noted https://mikyab.net/%D7%A9%D7%95%D7%AA/%D7%97%D7%9E%D7%95%D7%A8%D7%95-%D7%A9%D7%9C-%D7%91%D7%95%D7%A8%D7%99%D7%93%D7%90%D7%9F-%D7%93%D7%98%D7%A8%D7%9E%D7%A0%D7%99%D7%96%D7%9D-%D7%95%D7%90%D7%91%D7%95%D7%9C%D7%95%D7%A6%D7%99%D7%94 that in fact a decision is still required (a choice?) about with what force to throw the die and so on, so the question remains exactly where it was.

A deterministic creature has no option of chance, and therefore it has to choose between two symmetrical possibilities. A human being with free choice can decide to cast lots.

What good will chance do?
If it is meant to decide between the two symmetrical options, then now the dilemma is whether to cast lots, and that dilemma is not symmetrical. If so, the whole sting of the story is gone: man is not superior to the donkey in the ability to decide between two symmetrical options; rather, he simply recognizes the possibilities before him and knows that the discussion can be focused on a different question—whether to cast lots—which is not symmetrical…

Rabbi Michi,
now I understand the exchange here. Your answer explains what a person can do in a situation of symmetry, but we (the one who started the thread and I) are trying to understand the concept of “breaking symmetry.”
Casting lots is not breaking symmetry, but moving to a different situation that is not symmetrical.
If so, the donkey parable adds nothing to the question of free will. It does not prove that we feel freedom, because in this story the person will cast lots only because he knows the facts and knows that there is an option to cast lots (and that too could be done deterministically), and it does not explain what freedom is at all, because this case is not breaking symmetry but moving to an asymmetrical state, as explained.

I’ve given up.

It’s amusing to see how elegantly you avoid admitting your mistakes. Good luck going forward.