overdriving the next device in the audio chain ... sound will be bright, edgy, hard
No. What happens is that the signal completely saturates the input of the next stage. This usually causes instability as the amp oscillates in and out of complete saturation. Usually it sounds all muffled and blurred. The more the signal is saturated the more you get dropouts. Eventually total saturation of the input often produces no output at all as the device just sits on one of the power rails (flat lines).
Severely overdriving a device can also damage it permanently.
Anyway, I think that my explanation fits your observations. And let's face it, your observation do run counter to what almost everyone reports about the DAC1. (Never heard it called muffled. Too bright. Too clinical. Too harsh and other comments along these lines but never muffled.)
Perhaps your DAC1 was faulty?