This is a bit of a discussion for a different topic. In the context of this thread, nothing much has been measured at all about the comparator, which is one of the points being criticized.
However, in general the frequency domain and the time domain are interchangeable. That's why you can do a Fast Fourier Transform (FFT) on your data, follow up with an inverse FFT and - apart from tiny errors due to numerical limitations - end up with an identical signal in the time domain.
But there's a but: FFTs do use window functions, so their time resolution is limited. And all the graphs we typically see are from relatively long, time averaged FFTs over seconds and sometimes maybe minutes. That increases the SNR, which is desired. However, if you assume that a device has a hypothetical small glitch right at the beginning of the measurement, that glitch would be "averaged out" the longer you capture data for your individual time-averaged measurement. That's just how it is and is technically not even related to the fact that the frequency domain is being analyzed.
In addition, you do need to see the phase response on top of the frequency response to get the full picture in the frequency domain. That is rarely shown here on ASR. I would assume that's because it is very uneventful most of the time. If you analyze in the time domain and just measure the impulse response, you'll get the full transfer function of the device in a single plot which is very convenient. I'm sure that some types of defects are easier to spot in one domain than in the other, so there is value in both paths of analyis.
But now back to proposed audible differences in DACs!