It is all about conservation of energy. A piano is an almost rigid system that couples key, whippen, jack, lever, hammer. Pressing a key is the only energy input into the system. An oversimplified model assumes the system behaves the same way regardless of the energy input. In reality, the behaviour of several parameters (e.g., friction, effect of the escapement, compression of the felts) does depend on the energy input.
If we assume the key is pressed the same way, the initial energy input will be the same, regardless of hammer blow distance and other variables. The initial energy is a function of the acceleration of the key and is translated to a force (Fkey). This force is transferred to the whippen-jack-lever mechanism and finally transferred to the hammer as force Fhammer. Note: all these movements are not linear but rotational/angular since the energy transfer relies on levers that rotate around pivots. In any case, for a given initial force Fkey we end up with a force Fhammer.
Let's now add the "blow distance" to the system. This is the distance between the hammer (at rest) and the strings. The distance will impact how long it takes for the hammer to reach the strings and therefore the velocity of the hammer. If the blow distance is zero, then the strings are touching the hammer (the hammer would work as a lever lifting the strings). In this case, the velocity of the hammer will be zero. But the force applied to the strings is not zero. Fhammer would be a direct function of mass and acceleration (with velocity = 0). If we have a sufficient blow distance, the hammer will have time to fully convert acceleration to velocity. Fhammer would be a function of mass and velocity (F = m x dv/dt, with acceleration = 0). If we are in an intermediate scenario, then the hammer will still be accelerating when it hits the strings at a given velocity. In this case (acceleration > 0 and velocity > 0). So, part of the force will be a function of the hammer velocity and the rest a function of the acceleration. The force exerted by the hammer on the strings is the same regardless of blow distance, otherwise the system would violate conservation of energy.
So, why is the sound different depending on blow distance given that the force is the same? The main reason is hammer nonlinearity, i.e., the very complex way the hammer (felt) interacts with the strings. The simplest scenario is that of a completely rigid/static hammer. In this case, a ppp and fff sound would have the same waveform with a different amplitude. Another scenario is that where the hammer is considered to behave like a linear spring (according to Hooke's law). In this case the hammer compression is linearly proportional to the force. But the hammers in a piano behave non-linearly and compress differently at different velocities. If the hammer is travelling at low velocity, the felt will slowly compress against the strings while the energy is being transferred. If it is travelling at a higher velocity, the felt becomes rigid faster. The higher the rigidity, the faster the hammer will bounce back from the strings, which decreases the effect of damping the vibrations. In short, at higher velocities the hammer behaves as a hard (non-linear) spring that bounces back away from the strings fast. At lower velocity, a hammer behaves like a soft (non-linear) string that will remain in contact with the strings for a longer time. This effect is also responsible for changing the harmonics and therefore the tone.
So, the main function of reducing the blow distance is not to decrease the overall loudness but to the increases the control over the low spectrum of the dynamic range (say, pppp-p) at the expense of the higher range. Afaik, this pedal is also called "Mozart pedal" or "pianissimo pedal" and was at least available in some old Steingraeber pianos.