Does MTG have a card with a mana-generating ability that has a target and is thus not a mana ability?
Yes, Deathrite Shaman is one such card.  One of its abilities: 


  {T}: Exile target land card from a graveyard. Add one mana of any color.


In fact, its oracle page even addresses this ability:


  Because the first ability requires a target, it is not a mana ability. It uses the stack and can be responded to. (2016-06-08)