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> But apparently the constraints are: every player rolls one die once.

That doesn't eliminate the natural scheme, it just adds some useless die rolls. You can have each of two players flip a coin and choose player A when player A flips heads or player B when player A flips tails.

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Useless depends on context here; I believe the idea is that each player gets a feeling of agency (they go in the order dictated by "their" luck, not leaving the decision to someone elses roll).

The most pragmatic would probably be to shuffle N cards from ace(one) through N - and let everyone draw).

But that doesn't involve dice.

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