Don't forget that ab is a preposition as well as a signifier of agent. There is also no passive in this sentence for there to be an accompanying agent. Your translation is almost accurate, but change the "but" to a "from".
'Tantalus, being thirsty, desired to touch the river rushing from his mouth.'
As for why it is not an ablative, I believe you are talking about the ablative of separation? In that case, it is mostly used to indicate distance and not actual movement, whereas ex and ab do.
flebile nescio quid queritur lyra, flebile lingua murmurat exanimis, respondent flebile ripae