Thanks for the info!
Voltronic, my experience matches everything you've said. I'm pretty sure that what is holding me back at the moment is my recorder's self-noise (pre-amp and/or ADC), and sometimes the CA-11s self-noise. Now, for 90% of the shows I record (ranging from amplified acoustic music to very loud shoegaze bands), I have zero complaints. My best sounding recordings are probably reasonably quiet acoustic (but still amplified) ones without a lot of dynamic range. I use an old CA STC-9000 pre-amp into either a Roland R-05 or an old Zoom H1, and even the Zoom H1 has rarely been a problem, as long as I don't have to increase the volume by more than 10-15 dB in post. I know 10-15 dB sounds like a lot, but even an old recorder with a poor pre-amp like the Zoom H1 sounds more than adequate enough for this.
Where I run into problems is when (1) for some reason the performance is not amplified or just barely amplified, or (2) I have to deal with a lot of dynamic range. When dealing with (1), I can tell the noise is a mix of ambient noise, pre-amp/ADC noise, and perhaps microphone self-noise (I'm not sure I can tell the last one apart from pre-amp noise). I'm pretty sure I would get cleaner recordings with more sensitive microphones and better pre-amps, but the recordings don't actually sound too bad. They are listenable. The biggest issue isn't even the gear's self-noise, but that you can sometimes hear people around you breathing.
It's with (2) that I really struggle. When a 2h show has a few very loud peaks totalling less than 5 minutes, and I don't know when they're coming, I tend to keep the other reasonably loud parts between -15 and -20 dBFS (in my experience, the peaks will reach -3 dBFS or so). In a situation like this, if the quietest parts don't get quieter than -30 dBFS, I can boost them by up to 15 dB in post, and the end result will still sound pretty transparent. The noise won't be distracting. It's when the quiet parts approach -40 dBFS that the pre-amp noise really becomes an issue, even if the music is a bit louder in the room than in (1). You're right that this isn't the room's ambient noise, they sound distinctly different.
It's for (2) that I'm interested in a 32-bit float recorder. If I could get 10 or even 15 dB less self-noise for those quiet parts, it would make an enormous difference.