As you evidently realize, it is perilous to look for passive meanings in deponent verbs, since the whole point of being a "deponent" verb is that it can be characterized as "laying/putting" (pōnens) "aside/down" (dē) its passive meaning. It is true that deponent verbs do infrequently have passive meanings, but this is usually just encountered with participles, where things can get tricky. Since potīrētur is not a participle, we should consider it as having an "active" meaning and look elsewhere for the problem.
Here the problem is the common error of assuming that a noun in the ablative case representing a thing must be an ablative of means and so translating hōc vellere as "by this fleece." It is a good discipline to think of the ablative of means as not meaningful in itself, but simply extending the meaning of a verbal action where means is always understood, but not always worth mentioning. That means that verbs that already govern the ablative case for another purpose or that can take an adjunct with a different type of ablative (e.g., verbs of motion or verbs of separation) are not easily compatible with an ablative of means. Potior is such a verb.
Potior regularly governs its "direct object" in the ablative case, like ūtor, fruor, and fungor. The original sense of these verbs was probably something like "find use, enjoyment, mastery for oneself by means of or from something." From a Latin point of view, the arguments of such verbs can be understood as representing an ablative of means or source; however, from an English point of view, these arguments are simple direct objects.
Potior can also govern the genitive, presumably with the underlying meaning of "gain possession of"; however, that is not the case here.
With this understanding, we can see that vellere is just the argument, or "direct object," of potior. We can translate the last clause as "to obtain this fleece."
One other minor issue is that negotium is not always "a bit of business," although that is an ingenious way of finding a literal translation that works. The word negotium seems to come from nec (“not”) + ōtium (“leisure/idleness”). One thread of meaning veers toward "business, employment, affair," but another veers toward "labor, pains, trouble, etc." The expression negotium dare means something like "set a task to someone."
If we look at the whole sentence, it is something like:
"(What happened next was that) Pelias resolved to set Jason the task of obtaining this fleece." The ut clause is like purpose clause used to explain what the task was.
