> There's no such thing as an individual conscious self that persists over time

True. But then again, there is nothing that persists over time. Entities with enduring identities - of any kind - are just abstractions that we superimpose on experience.

> Consciousness is just something that living beings do

To my eyes, you're switching over to another meaning of "consciousness" here. Sure there's no enduring self, but that doesn't mean consciousness (the capacity for experience, rather than mere behavior) is just something we do. We can understand feelings, thoughts, emotions etc as fundamentally "impersonal", yes, but that doesn't mean that they are not states of being. To me such states are about as real as anything. Again, it's two separate issues: 1) the nature of a persistent self, 2) the nature of mental states, not taken as "possessions" of such a self.

(Still, psychologically speaking, the sense of self is baked into even our most basic acts of cognition. When you see an apple, there is always an implicit "you" in relation to the apple. In practical terms, it takes a lot of effort to separate one from the other - yet another topic!)