Given our current and best information about the physical world, unless I am missing something, I don't see how consciousness as well as the subjective experience that forms from it, can't be safely explained as a purely computational phenomenon.
Take the simplest of computational networks - two states going through a logic gate, producing a new state. According to the research that I am aware of, examples of which I write in the next paragraphs, this simple network, by itself, can be regarded as a fundamental level of consciousness, or a single block of logic if you will. If, for example, you want it to contain memory, in order to process that memory and produce a new state, two NOR gates will suffice. Connect them with another gate and a binary sensor and you essentially have stored information processing which also depends on the environment.
There is already an amount of research around programming microorganism behavior with a combination of logic gates - which is the fundamental computational mechanism in electronics. Example nice reads:
https://www.sciencedirect.com/science/article/pii/S030326472200003X
https://arstechnica.com/science/2010/12/building-logic-gates-with-bacterial-colonies/
Beyond that, we are just describing different complexity levels of "logic". From what I understand molecular neurotransmitter function (that mostly work as emotional regulators in humans), can be boiled down to logic gates as well. For one, they seem to work similarly to AI neural network learning algorithm techniques to encourage or discourage decisions by altering neuron firing frequency, and even if one could argue that neurotransmitter effect on neurons is not binary, unlike logic gates, their analog behavior can be replicated with binary behavior. Again, by looking at something we can actually map, neurotransmitters in earthworms for example, work in their nervous system as a decision regulator.
By taking a look at the animal kingdom to comprehend our "seemingly inexplicable phenomenon" of consciousness, we can see that the more complex this network of logic is, the more behaviors emerge from it. In vastly more complex social organisms like bees, research has shown that they share more "traditionally human" behaviors than was thought before. Some name that level of complexity "sentience" - but what does this sentience describe, if not something that just describes a greater level of similarity to our own "special" experience, and not something unique or a separate phenomenon.
In essence a decently complex lifeform, is self-powered, has sensors that constantly gather information from the environment, can store an amount of memory, and contains a mindbogglingly complex neural network regulated by neurotransmitters that makes decisions.
Moving on to more complex lifeforms, their similarities to our species increase. There are important differences, for example, like the capacity to store long-term memory, or the evolution of a dedicated neurotransmitter network (Amygdala) and many more, but at the end of the day, it boils down to the aggregation of complex computational systems.
As "the hard problem of the consciousness" in the sense of how "gives rise to subjective experience", I don't see how it's not just simply a subsequent symptom of the complexity of our systems and the randomization of information. Randomization of information exists in every aspect of our conscious being. From our imperfect sensory inputs, to the wiring of our neural networks and the unique set of experiences and DNA that helps it form.
Beyond information randomization, in theory, the quantum mind hypothesis could further explain and bridge the probabilistic nature of cognition that gives rise to subjectivity, but again, this is well within the realm of soon-to-be conventional computation. Anyhow, I think that speculating or even philosophizing around this kind of a black box is counter-productive to the discussion, so I won't touch it further.
If there is information that dispels this, please, go forth and explain.