Two people far more qualified than I just answered while I was typing this, so much of the below is a rehash of what they said worded slightly differently, but I'll post anyway. My numbers my be off slightly but the basic idea is sound-
Your basic concept is correct. More bits = more dynamic range. A 24 bit file can encode 144dB of range but not even the very best analog equipment is capable of 144dB. The limit is the analog circuitry in the recording chain before the signal is digitized. Good studio equipment can manage about 20bits of usable dynamic range. The relatively inexpensive gear most of us use around here is probably closer 18bits when set to write 24bit files, most of it probably closer to 17bits, some no better than 16. The 24bit files will simply have multiple bits of random analog noise at the bottom.
Dynamic range is limited by clipping distortion at the top and dither noise at the bottom. The noise at the bottom is not quantization noise in any modern digital recorder, it is dither, which is more benign and sounds like analog hiss (because it is) and which is used to eliminate any quantization noise which is not benign and sounds terrible. But that dither noise is going to be much lower than the analog noise of the circuitry in almost all cases, and most of the time, that analog circuitry noise is going to be lower than the ambient noise of the environments in which we record.
So in the 'real world' of concert recording, if you figure out how to set things properly you can do pretty much exactly as you suggest. I now do that myself with one of my setups-
I still adjust the gain to peak at about -15 or -18dbFBS at the start and leave it there without worries when I'm running mics and recorders onstage or whatever, but with my worn surround rig, I've figured out where to set things so that I can leave the gain set the same for everything I record, partly because it's a pain to match the gains across all 4 channels evenly if I needed to change it. Instead I leave it set low enough to handle the loudest stuff, and let the quiet stuff peak lower than most people would want it, then amplify it digitally later. Even for very, very quiet music in very quiet rooms, the ambient noise floor of the room is higher than the analog noise floor of my recording equipment which is maybe getting about 17-18dB of range if I'm lucky (but I'm guessing about that). And I'm talking things like certain sections of classical material with sound levels hovering around the threshold of hearing in a couple halls designed for performance with quiet HVAC and isolation from outside noises. Some of those recordings peak at best around -30dB, which used to freak me out, but they sound fine once amplified. I've only noticed the analog noise on super quiet nature recordings using this setup, which isn't it's intended purpose anyway.