I don't deny that weight is a significant factor in the cost of a flight. But I'd want to contract a flight from Buenos Aires to Orlando (Disneyland) on a Samoa Air 747, and I'd fill it with 400 5 y/o children and 40 adults taking care of them, and I'd like to see if Mr Langton still things that paying for the kilo is the fairest fare. With 100% of the seats occupied, the plane will be taking like 1/5 or less of weight (and revenue) in passengers and baggage.
The truth is:
Only half of the drag is related to weight and hence to fuel consumption.
Depending on the type of plane and distance flown, only 25% of that weight is payload. The rest is empty weight and fuel.
So the fuel bill of a zero-payload flight can be say 12% lower than the fuel bill of the full-payload flight.
The other 88% of the fuel bill and 100% of the rest of the costs is weight-independent (crew salary, maintenance, airport fees, leasing costs).
On the other side of the equation we have revenue. If the payload is limited by the max weight that the plane can carry in a particular flight and you had the ability to sell more payload than that, then all the payload taken should pay the full bill, meaning that fewer fat passengers should pay more (each) than many thin passengers to keep the same total. In this case, the wight of the airplane and of the payload is fixed, the cost of the flight is fixed, so it would be fare to charge by the kilo to keep the revenue and hence the operative earning of the flight also fixed. But please note that we are assuming that you had the ability to sell more kilos of payload than what the plane could take. If not, then the revenue is not a function of the weight, and things like my first example could happen.