A good question. But hampered, as are many such discussions by mismatched terminology. There is no Buddhist concept that quite matches the idea of "consciousness". We don't really notice this because the early translators used quite imprecise terminology and we're stuck with it. If there is one thing that vijñāna is not, it is "consciousness".
Another factor that one must take into account is that what ancient Buddhists meant by "mind" and what modern philosophers, like David "The Hard problem Chalmers mean by it are two different things. Buddhists for example make no categorical distinction between thought and emotion. There is no word for emotion in either Pāḷi or Sanskrit. See Emotions in Buddhism. Emotion is a kind of citta, as it thought, or perception. And yet the distinction between thoughts and emotions is critical to modern philosophy.
Nor is there any notion of the mind as a container for experience. The metaphor of the "theatre of consciousness" is entirely absent from Buddhist thought. See The 'Mind as Container' Metaphor.
The early Buddhists never claimed to fully explain conscious thought. The project would not have interested them. They were trying to understand dukkha - unenlightened experience. They were interested in the intoxication (pamāda) of ordinary people with sense experience and how to sober up. However as I understand it they would probably have agreed that because experience arises in dependence on sense object, sense faculty and sense cognition, and that most of us proceed without pause to proliferation, that a complete explanation of a conscious thought would be difficult. But I don't think anattā has anything to do with this.
On the other hand experience of samādhi changes the ball game. One can get a kind of insight into the arising of cittas that are as yet not incorporated into the modern theories of mind.
Consciousness is only explicable by way of an explanation of some particular conscious thought.
The trouble is that what you are calling "consciousness" is almost certainly ruled out in Buddhist thought by anattā. From the early Buddhist point of view there is no such thing as "consciousness" in the sense that, say, David Chalmers uses it. There's just conscious experience. Chalmers own solution that "consciousness" must be different from matter is also ruled out - if something is not accessible to the senses then we can have no knowledge of it. And this is the most important implication of anattā.
It does leave open the question of the Hard Problem though, because there is still something that it is like to have an experience. It's just that solving this problem would not tell us anything interesting about why we suffer. We suffer because we have an aberrant relationship to pleasure/pain.
So it seems to me that you're barking up the wrong philosophical tree with this question. However, it's worth asking because it helps to clarify the distinctions between Buddhism and philosophy.