There are other impurities than air, but they don't contribute making the ice cloudy. The only thing making ice cloudy is the bubbles. And you can verify it experimentally in your own freezer just by using a different freezing technique to create perfectly clear ice: https://youtu.be/E1G1I1LJjrI
I don't think this video is really sufficient to prove your point that "the only thing making ice cloudy is the bubbles". The directional freezing technique is essentially a filtration method - the H2O freezes first into the growing crystalline structure and all the other impurities (various minerals AND dissolved air) get "pushed down" and concentrated into the liquid water below.
As evidence that directional freezing can filter out impurities other than air, I would point out that the directional freezing process is an effective enough filtration method to filter salt out of sea water in sea ice. [1] So my assumption is that those other minerals commonly found in water (lime, calcium, fluoride, magnesium, etc) are also being filtered out in the process - and it certainly seems plausible to me that filtering those out contributes to the clarity of the resulting ice.
So my intuitive understanding, along with what I could learn with a bit of googling, is that there are a number of impurities which can make ice appear cloudy including (but not limited to) dissolved air. A number of people here seem to be adamant that it's just air bubbles, so I'd honestly like to know why they believe that to be true (e.g. sources, some clear explanation, etc)? So far the only justification I've seen is "because directional freezing works" - but as explained above I really don't see that as being a sufficient justification since it filters out a number of impurities other than air.
