Thank you for your replies! I must be repeating countless discussions about consciousness that already happened on this forum. I don't intend to keep this going for much longer. Maybe it'll be my post on this topic.
Could we reword the claim "consciousness is an illusion" as "consciousness is created by neural activity in the brain"? This should address the questions about how the consciousness is borne. Let's say we make this claim because we can see the correlation between neural activities and consciousness: no activity means no consciousness, and altered activities correlate with altered states of consciousness.
In saying the above, we're also saying that nothing other than neural activity in the brain gives birth to consciousness.
If we ask a conscious individual about how they perceive their consciousness, they are likely to say that consciousness just exists unconditionally (like god). We don't perceive consciousness as the firing of billions of neurons.
In other words, the inner workings of consciousness don't seem to accurately represent the physical reality that gives birth to consciousness (neural activity in the brain). This crease a basis for us to suggest that "consciousness is an illusion" (i.e. a deceived appearance), meaning that the way we perceive consciousness does not accurately represent the physical reality of how the consciousness is being borne.
I hope you can follow my train of thoughts, even if you already see some flaws in it. With everything above considered, when you say that "consciousness is an illusion" is necessarily wrong, I perceive it is as a claim that consciousness is
not the result of neural activity in the brain. This means that consciousness exists unconditionally. I can see some merit in that, but I'm also not fully satisfied by it. I can see it as a form of Curry's paradox: "If consciousness exists, it exists unconditionally", which means that "consciousness exists unconditionally".
I can agree that the statement "consciousness is an illusion" is somewhat paradoxical, but I don't think that denying the statement straight away is a fair way to end a discussion on this topic. I think a better way is to offer a more elaborate and less contradictory way to think about consciousness.