I think Gandalf's words at the Battle of Five Armies - '... Bolg of the North is coming... The bats are above his army... They (I take this to refer to 'his army' and not 'the bats') ride upon wolves and Wargs are in their train!' - rather counteract the impression you put forward.
My take on it has been that the Goblins of the central Misty Mountains (Hoarwell down to Moria, more or less) sometimes ride Wargs, as you suggest; but further north, around the Ettinmoors/Mount Gundabad/Mts of Angmar/Grey Mts, and perhaps in the wide lands even to the north of them, the Goblins ride wolves in large cavalry armies, and Wargs are more often a force allied to the Goblins, rather than ridden by them (for mutual gain or not). Hence, at the Battle of Five Armies, the 'wild Wolves' (which I presume = Wargs as it's capitalised, and therefore not the same as the 'wolves' that the Goblins ride - which may be considered tame) are a force in their own right.