That's like saying movies technically don't exist because they're just a series of still frames. That sequence of still frames is a real thing in its own right, and it has a name.
The movie is more analogous to the conscious person, or perhaps “the mind,” and no one is proposing that those things don’t exist.
Insisting that consciousness must be real because it feels undeniably real is analogous to insisting that the way movies work is by showing continuous motion because that’s undeniably what it feels like when you’re watching a movie.
The difference would be that in the movie case, there is a you that is being tricked, that is present even through the trickery. You're having the experience of something moving, but it's really your senses being tricked into constructing this experience out of something else.
But for the case of consciousness, the "what's being tricked" is having experience at all. So the claim "insisting that consciousness is real because it feels real" rests on the cogito-ergo-sum like observation that if there were no experience, there would be nobody to feel it was real to begin with because feeling is an experience. That you are feeling it (in a first person sense) is evidence in itself, although frustratingly, not evidence that can be communicated.
That's what makes it so hard, though. Either "cogito ergo sum" seems self-evidently true, or it seems self-evidently false, and there's no way of getting to the position by indirect means. And because it can't be communicated, there's seemingly no way to unambiguously show someone else what you mean.
