OdBFS shouldn't be a problem for ANY converter. It's when you have inter-sample peaks that the distortion becomes a problem. Most "good" converters can handle these peaks really well. It's the poor designs and the ones where the manufacturer has cut corners to meet a certain price point where hot signals become a problem.
I really wish we could meet with a white board and maybe a beer; I suspect you know all I am about to say and we could skip the board. I started to try to explain but it got too long and deep. There's no such thing as "inter sample peaks" over time; some peak will exceed the dynamic range at the sample point eventually. If properly filtered, you can't get a peak over about 1 lsb over FS without clipping, and then 0 dBFS no longer applies -- it's +1 dBFS or whatever. Since we are talking DACs, not ADCs, there's no way to generate peaks, inter-sample or not, greater than 0 dBFS at the DAC's output. Well, for a perfect DAC, and in the long term...
Here's what I believe, and have measured, and you can undoubtedly go to ADI and other web sites for more information: DACs glitch. Real-world designs have overshoot, ringing, and output glitches that exceed 0 dBFS, sometimes significantly. Those signals can overdrive and saturate the output buffer amplifier, causing all sorts of nasty distortion. I suspect that is what you are hearing with "hot" signals; the DAC itself may be part of the problem by generating the glitches, but the amplifier stage(s) after it muck up the signal by saturating and not recovering quickly.
Unless you are speaking of "DAC" as most audiophiles use the term, including the preprocessing, actual digital-to-analog converter, and output filters and buffer amps, all as "the DAC".
Most converters (ADCs and DACs) are tested at -1 dBFS. For various reasons performance tends to fall a little right at full-scale.
Your last line is spot-on, and of course applies to many things in many fields... - Don