I feel like I am conscious. If consciousness doesn't exist, then there must be some sort of illusion of consciousness to make me feel like it does. I see no reason not to call this illusion of consciousness "consciousness", since it fully explains what we mean when we say "consciousness".

Therefore, consciousness exists.

This sounds like some sort of horrible misuse of Dark Arts, and it would be if you plug in any other term (for example, "God"), but since "consciousness" has no properties except "a certain experience people perceive themselves as having", whatever it is that makes them perceive that experience satisfies the definition of "consciousness".

This is not an explanation of consciousness or at all helpful in the formal study of consciousness, but it's why I don't think "consciousness doesn't exist" solves anything or is particularly useful. Okay, fine, it doesn't exist, now there's still exactly the same amount of mysterious experience to explain as before.