This is admittedly a late answer. Hopefully it will clear up some of the confusion.
If you ignore aerodynamic drag, ignore that the coefficient of rolling friction varies with load, and ignore a number of other factors such as friction between the axle and the bearings that support it, then yes, stopping time / stopping distance is independent of vehicle load. This is completely unrealistic reasoning.
Things get a bit more complex if you don't ignore those factors. Consider a pickup with or without a load of dirt. I'll start by assuming a constant coefficient of drag and by ignoring everything but aerodynamic drag. The only horizontal force acting on the vehicle is drag, and that is strictly proportional to velocity. Per these assumptions, a fully loaded pickup and empty pickup moving at the same speed suffer exactly the same drag force. Since acceleration is inversely proportional to mass, the loaded pickup takes longer to come to rest than the empty pickup.
Most people drive their pickups with the tailgate closed and the bed uncovered. This makes for a rather non-aerodynamic vehicle. Some people solve this by replacing the tailgate with an open mesh net, others solve this by adding a bed cover. I'll assume a typical pickup owner who hasn't done either. Filling the bed with dirt makes the vehicle more aerodynamic. This decrease in the coefficient of drag will make the loaded pickup take even longer to come to a stop than the simple analysis above.
The above ignores friction between the tires and the road. There will be a significant increase in contact area between tire and road after filling the pickup with dirt. This increased contact area increases the coefficient of rolling friction. This will act to make the loaded vehicle come to rest more quickly than the empty vehicle. Whether the loaded pickup has a longer or shorter stopping distance than the empty truck depends on which of aerodynamic drag or rolling friction predominate.
Anyone who has hauled a load of dirt with a pickup should know they need to reinflate the tires after taking on that load. Driving with a heavy load and tires squished flat to the ground is a good recipe for trouble. Reinflating the tires so that the contact area is more or less the same as it was prior to taking on the load reduces the coefficient of rolling friction. This makes drag and rolling friction work in concert with one another. The loaded truck takes a good deal more time or distance to come to a stop than does the empty truck.
Similar concerns sometimes come into play during the launch of a rocket. Rockets with a high thrust to weight ratio oftentimes need to coast for a while between the shutdown of one stage and the ignition of the next. When that's the case, the separation of the burnt-out stage doesn't occur when the stage shutdowns. That separation instead is delayed until shortly before the next stage is supposed to ignite. Keeping the burnt-out stage attached until the last minute reduces drag acceleration by a considerable margin.