Let's say for simplicity's sake that, without adjusting the DAC's volume, if you were to cut out the loudest section of a song, you would get a signal of 2Vrms. Likewise, let's say that the quietest section of the song is a couple moments of complete silence, such that if you were to cut the silent section of the song out, you would get a signal of 0Vrms.
Now, reduce the output volume of the DAC, and the loudest section now boasts a signal strength of 1.5Vrms, but the lowest section of the signal is still 0Vrms, since it can't really be made quieter.
Again please correct me if I'm wrong, but it seems that the relationship between the highest and lowest part of the signal has changed when affecting output volume, no?
problem 1:
no sound isn't a level of sound. I get your idea, except that it relates to nothing in real life. not for a DAC, not for sounds in the air. the concept of silence is in fact a lot of noises, just a lot quieter than the music. so the only concept where your rational works is the concept of Signal to Noise Ratio(SNR). your playback system will have noises, even the DAC alone will generate some, and when you lower the voltage output to reduce the listening level, some noises will go down along with the music, but some will not. instead they will remain just a noisy as ever. so the difference in amplitude between the music and the noise will effectively be smaller.
problem2:
but for the music, in practice a song will pretty much never be silent, when it is, the song typically hasn't started or has already ended. if you look at the silent parts within a song, you'll notice that they're usually only 60 or maybe 70dB below full scale signal. I'm sure there is somewhere a song with no signal within the song, but I wouldn't count that a the norm because it really is not.
so just with that fact, let's look at your idea. a DAC will have let's say 100dB of dynamic range and SNR(on sigma delta DACs it's usually the same value). you have a song that effectively uses 60dB of dynamic and that's what you'll get at the output. it's not like the DAC with 100dB of dynamic will somehow stretch the song so that it now becomes a song with 100dB of dynamic. the dynamic range of the DAC is what it will use with all songs. maybe the terms are misleading because dynamic is in them?
now if we take the case of lowering the voltage output(AKA listening level) by half. so instead of a rather typical 2V, the DAC will now output a full scale signal as 1V. that's only -6dB. so you'd need to reduce the voltage by a good deal more for that reduction+ the 60dB of our previous song, to amount to more than 100dB that the DAC can handle. and even when we reach that limit, what will happen is that the quieter signals will be lost, even then the DAC will not dynamically
compress the music. the song will effectively have lower dynamic range, but because the quiet parts are gone. it's something pretty specific.
also modern DACs may have a specif way to handle volume attenuation that doesn't risk crippling the signal so nothing would actually happen beside having quieter music.
problem3: if you wish to consider that actual silence or at least the quietest signal a playback system and a room can let you perceive, is part of the song, then it would mean that you'd need to play the song at a specific listening level to get the "correct" experience and "dynamic". but what would that be? I have no idea and don't know of any such information on a record. and if noises from your street become louder, should you then increase the listening level in proportion to maintain the "silence" at the right level relatively to the music? or the opposite, if you play the music louder than it should be, you'd get a silence that's too quiet relatively to the all dynamic of the song ^_^. this is more of a philosophical question than anything usable but I get that you'd ask yourself that, because I did the same a few years ago