Exactly, you have a handwave. A vague idea that when a system becomes "complex" enough it suddenly becomes conscious. There's no mechanism, no rigor, no cause and effect. With the browser you can chase down every bit through the hardware, down to the quantum tunneling effect dictating what's happening in the silicon and come up with a concrete explanation based in physical laws.

With the browser example, at no level of abstraction does there exist any ambiguity. "Emergent" in that framing simply means we can explain abstracted behaviors in terms of other abstractions. But if you wanted you could drill down to the particle physics and have a complete, rational explanation of the system.

Not so with consciousness, and thus the "hard problem"