Freight packed into a container (be it a plane's cargo hold or truck or something else) can be said to either "weigh out" or "cube out":
- Weigh out - The weight of the cargo is equal to the maximum carrying capacity, even if there is more space available. Example: a truckload of gold bars
- Cube out - The volume of the cargo is equal to the maximum carrying capacity, even if the vehicle can handle more weight. Example: a truckload of balloons
An airline's dispatcher will calculate a weight limit for a given flight based on the aircraft, the route, expected winds and weather, fuel needs, and other factors. A long-haul flight may need large amounts of fuel, reducing the weight available for cargo and making it more likely for weight restrictions to be applied, while a short-haul flight may be more likely to cube out, as the cargo space can be filled up to its volumetric capacity. If a flight happens to cube out, checking a gigantic empty suitcase may still cost the airline money, as you're occupying volume that could be used for other paying cargo.
As such, airlines are interested in both the weight and volume of cargo to maximize the amount they can carry, or at least price for it proportionally. In the cargo shipping world, this is often handled by dimensional weight: the shipper essentially pays extra to ship large, low density cargo. This is the process the industry uses to best recover their costs for shipping cargo. However, For checked bags, the less rigorous standard of linear inches is used, presumably because the average density of suitcases is fairly standard, and because enforcing such a policy would be overly difficult: passengers need something predictable and easy to understand. As such, measuring linear inches reasonably captures the major cost drivers for air cargo while providing a policy that can be reasonably implemented at check-in counters.
Speculating idly, it is also clear that measuring volume is also more difficult at the check-in counter. Suppose one has a 30" x 19" x 12" bag, giving a volume of 6,840". A measurement error of just 1" on one side could lead to a volume measurement of 7,410", an 8% increase. A rule that must be quickly enforced by airline clerks at the counter should not multiply small errors into ones that could quickly spell the difference between an allowed and a prohibited bag. And dealing with anything non-rectangular requires even more rules. In contrast, a 1" error when measuring the linear dimensions of the bag will have a small impact on the acceptability of the bag.
If a bag is unusual in size or type, it may not be able to go through the standard conveyor system (even a cardboard box is sometimes treated specially in my experience). But my understanding is that these size policies are primarily about limiting the amount of volume the bags take up in the cargo hold, not their processing through the airport's baggage system. Once it's in the cargo hold, they'll do their best to pack it in whether it's cube shaped or long and thin.
And that shouldn't be that surprising, because many airlines will take odd sized items for sports equipment like skies, sometimes at no extra charge depending on airline policy. There may be a physical limit for really long items though, such as this one (from United's policy):
Hang gliding equipment that is more than 72 inches (183 cm) in length
cannot be accommodated on 737 series aircraft. Hang gliding equipment
that is 108 inches (274.4 cm) in length or more cannot be accommodated
on Airbus A320 or Airbus A319 aircraft.
There's a point where something is just too long to fit in the cargo hold. They're obviously not going to take something that physically can't fit, which has to include making it around the corner into the cargo hold (cargo doors are actually pretty small, especially on narrowbody and regional aircraft). The linear dimensions help prevent that case. If they measured volume, some joker would try to show up with a 600 X 1 X 0.1" box and would have to be told that's not going to fit through the door (though I'd want to know how you got it to the check-in counter in the first place).