Well I think you can do it with a contest, but even if it requires an attack roll; that does not necessarily make it part of the attack action. There is a difference between an attack roll and the attack action. Your familiar example actually illustrates this point.
From the Find Familiar Spell:
"..... when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. If the spell requires an attack roll, you use your attack modifier for the roll."
So even though a familiar "can't attack" (and I agree with you on that) you can within the limits of the spell, roll an attack roll for it to arbitrate success or failure for something other than the attack action that it can do. Your argument so far is you can't use oil in this way because using it requires an attack roll. An attack roll is different than the attack action and pouring oil on someone is not the attack action, it is use an object action.
The familiar cant take the attack action and as such can't use its claws but as the spell wording clearly demonstrates it is able to make an attack roll for things other than the attack action.