its just that the momentum has to go somewhere eventually.
Watch out. Momentum conservation is not like a set of accounts that has to balance at the end of the tax year. It is balanced instantaneously, as a consequence of Newtons second and third laws.
Both are right.
Returning to the example of an item dropped from the ceiling of a plane - yes, the momentum transferred to it by force of gravity balances instantaneously, by third law. It balances by accelerating the item, in free fall, by second law.
But the item cannot get out of the plane. When it hits the floor and comes to stop, its momentum returns to zero. Thatīs when the entire weight and momentum goes back to the airframe.
So, when you count the momenta of moving parts, the momentum must be conserved instantly. But since internal momentum cannot be nonzero in long term, eventually the momentum must be conserved without any allowance for moving parts.