You're right that there's no
proof that consciousness can be explained by biological function alone. But my belief is that the preponderance of the evidence makes it the most likely explanation by far.
> You can demonstrate that humans react (and quite predictably) to weather - when it rains, they take their umbrellas out, when there's a hurricane, they seek for shelter, and so on. It's not evidence that human behaviour is an emergent property of weather and nothing else.
In the literature, there's a term "necessary and sufficient" which is used quite often. For instance, destroying a certain percentage of a certain type of dopamine receptors is necessary and sufficient to produce parkinsons-like symptoms in rats.
Rain can be demonstrated to make people take umbrellas out, but it's not sufficient. Sometimes it rains and people don't take umbrellas out. It's also not necessary, sometimes people take out umbrellas when it's too sunny. So it's hard to establish that strong causal relationship between rain and umbrellas.
With consciousness, we can't point to a single example of human consciousness which is not at the same place at the same time as a reasonably well-functioning human brain. We can't prove 100% that it's sufficient, but it certainly seems to be necessary, and there's zero evidence so far it's not sufficient.
> The big question persists: how does it happen that a computational system, no matter how complex, FEELS something?
This is actually fairly well understood. So if you are talking about "feeling" in the sense of emotion, we actually have a very detailed understanding of how that system works.
So some of our lower brain regions are responsible for preparing our body for action. For instance, if you've been bitten by a dog before, your brain might learn to get your body ready to run when there is a large dog around. Your sympathetic nervous system will kick in when your visual cortex detects the right patterns, and elevate your heart rate, quicken your breathing, and cause your hairs to stand on end.
So your emotions are essentially a sense which observes that type of sympathetic arousal in your body. Your emotional systems will notice that your body has entered flight-or-fight mode, and will interpret that based on your context and memories to signal to your higher brain function that you are experiencing fear or anxiety.
So that's just one example, but it's extremely plausible to me that consciousness/subjective experience is either just the sum total of all these functions the brain is performing, or is some emergent property on top of them, or else is even some highly specialized function of some subsystem of the brain we don't fully understand yet.
For instance there are credible arguments that the thalamus is the seat of consciousness in the brain.