tube DAC noise

I have no experience with the Black Ice to know if the noise is inherent, or simply that you have noisy tubes. Try swapping in some tubes that are rated low in microphonics

To your question if all tube DACs are noisy - I have a Modwright Elyse tube DAC and have zero noise issues

I own the Jolida version of your Black Ice DAC, and I have had no noise issues with it.
Originally, I did swap out the stock tubes with replacements . That, may have prevented me from experiencing the same problem as you have.
I own a tube preamp, and, after two years, it developed a hissing noise. I swapped out the stock tubes with new tubes, and that took care of the noise.

My experience is that tubes have noise.  Usually not circuit topology.

No, tubes are not noisy. I have all tube equipment on my main system and four headphone amps..; no noise... none. Total of nine components.
One can get a noisy tube occasionally. Sorry don’t know anything about black ice.