First: This very distinction turns against you, for if the contract is reclassified by his presence as one of reciprocal exchange—in fact, if two employers (jāʿilān) make a juʿāla (contingent-reward contract) and each pledges a reward to whoever performs a task equal to his own—the agreement is, by unanimous consent, lawful even without a muḥallil.
Second: Your position would oblige you to introduce a muḥallil and permit the joint stake in every other lawful pursuit: footraces, swimming, calligraphy, sewing, carpentry, and all other legitimate crafts. If the muḥallil turns the agreement in those cases into a valid juʿāla, why should he not do so here? What difference is there? The point is perfectly clear.
They also said: Moreover, the muḥallil’s participation must be either to make the prize lawful for himself or to make it lawful for someone else, and both suppositions are untenable.
As for the first, its invalidity is obvious, for he enters only on behalf of the two contestants so that, in your view, their agreement will not amount to gambling. The majority of those who stipulate his presence have stated explicitly that he does not join in order to claim the prize for himself, and they have weakened and dismissed the claim of anyone who says otherwise.
And they are right, for his supposed entitlement to the prize could rest only on one of two grounds: either that he finishes first, or that he did not contribute to the stake. If it is because he wins, then winning itself is what renders the prize permissible; thus whoever Allah (1) the Exalted blesses with coming first (2), the perfection of his happiness (3) is that he alone should enjoy his provision, and the muḥallil has no effect whatsoever.