If I'm watching a movie, and decide that the loud bits are getting to loud for the neighbors, DRC provides the fix by reducing the level of the peaks, while also increasing the level on the dips.
Applying DRC, and then having to turn the volume level down because everything except the peaks have been boosted, doesn't make sense. It is the peaks that are disturbing the neighbors, so boosting everything towards peak levels is counter intuitive. I should be able to apply DRC to not annoy the neighbors, without also having to reduce the volume level. Then at some stage after this, disable DRC, and have to change the volume level again.
I'm pretty sure that DRC was originally not intended to adjust the loudness (or average volume or whatever you want to call it).
Now lets look at it from a SNR point of view.
When you apply digital boost, you boost the noise by the exact same level as you boost the signal. It should be obvious why you don't want to apply large levels of digital boost.
When you reduce level, you have a negative effect on the transient response of the signal. It should also be obvious why this is an advantage in a DRC situation.
An excellent feature would be one similar to that applied by Yamaha, they call it adaptive DRC. The level of DRC is adaptive based on the volume level. This is an excellent feature, since it provides varying levels of DRC (including no DRC) based on the volume level without further user intervention. Generally, as you reduce volume level, you actually want some boost applied to help reduce the chance of the signal reducing below the noise floor of the environment, and, you want some reduction of the peaks, since this is probably why you were reducing the level in the first place, to reduce the level of the peaks. When volume level is +/- 0dB or greater, no DRC is applied.
edited for spelling.