At least in the United States, most people don't pump cold tires. They drive to the gas station and immediate pump. Yes, mechanics pump cold tires, but mechanics know, or have resources to look up, how much to pump; owners don't as easily. Why, then, do labels on cars list only the pressure to which to pump cold tires, and not the pressure to which to pump warm ones?
Because "cold" pressures are much more likely to be consistent then "warm" pressures and barring things like one side of the car having been in hot sun for a few hours should be pretty evenly "cold" all round as well.
Driving a mile or two to the garage to check pressures and inflate them is unlikely to have a significant impact on tire temperature so is generally considered "close enough", particularly for road cars which aren't going to be hugely sensitive to small variations in pressure anyway.
On the other hand if you have done a substantial drive to get there the factors affecting temperature/pressure will vary hugely, how much acceleration have you done? How much hard braking? (Heat dissipated from the front brakes can heat tires significantly - and usually much more so than rears) Have you cornered hard? Have you done hard cornering in one direction but not the other? Has one wheel gone through a deep puddle of cold water?
All of these things are big unknowns that make trying to pin down a figure for what a normal "warm" pressure is very difficult hence these are given as "cold" pressures from which the manufacturers can extrapolate that under "normal" usage the tires will be within their operating range.
My guess is that cold pressure is a more repeatable measure. Warm or Hot tire pressure can be skewed by the amount of moisture in the air inside the tire. If you take two tires, one with air with 75% relative humidity and one with 0% RH, then heat them up. The tire with moisture will generate higher pressure. The air in the two tires have a different coefficient of expansion.
And, if you measure your tires cold, the working pressure is going to be above the cold reading. If you measure your tires hot then working pressure is less easy to guesstimate and will likely be lower than the hot reading. Under-inflated tires are bad!
Also, one of the reasons for the change to Nitrogen in auto tires is that Nitrogen is less hydroscopic, i.e. it has less capacity to absorb water. And presumably will be less affected by the reading temperature.