All the filtering is in the DAC chip itself, unless it specifies otherwise. The buffer stage can have filtering but it is not because the DAC needs it.
Huh, I see. Of course DAC doesn't need it, but the U/I converter needs it. That is the reason the analog stage is picking all that noise as it must cover high frequency band and there's no filter in the buffer stage to cut it at Nyquist freq. IMO that is not how it should be done. If there would be a filter in the buffer stage that would cut after the Nyquist things would look much better with your measurements.
But, of course, doing everything in one chip and simplifying the analog stage cuts the cost.
Btw, why should we care about out of the band garbage when speakers/headphones anyhow can't reproduce them? Can they disturb the amps?