The issue with calling it "emergent" is there's a categorical gap between the phenomenon.
Perhaps we should start with a little set theory. You "philosophy" studying guys like to think of yourselves as knowing some math and reasoning, let's put it to use.
Start with the set of all things that have consciousness or mind (even if you think these are not quite correlated, just go with me, OK). Then the set of all things with brains.
The things with mind set is a sub-set of things with brains. All things with minds have brains, but not all things with brains have minds.
We can impact the mind by physical stimulation of the brain, etc. So far no minds have been found in anything without a living brain. Find one and we can talk about it.
This is strong circumstantial evidence that minds are an emergent property of brains and then it is up to the relevant researchers to investigate. (And no, they are not going to consider supernatural causation in their studies.)
It's not simply a more complex interaction, but an entirely new development.
There are all kinds of emergent phenomena that seem like "entirely new developments" when examined superficially.
It's analogous to claiming that you could arrange white blocks in such a way that they would produce a black tower.
I don't know about black from white, but I could definitely take a bunch of transparent stuff and make an opaque something.
This is a bad analogy anyway. With black and white you are talking about two different collective properties that are in the same category.
I could show the equations of atmospheric motion to an expert and they couldn't point to the term or equation that makes hurricanes occur, yet the equations do describe general atmospheric motion and when solved on a computer will generate computerized hurricanes without any prompting or additional input. The hurricane is an emergent phenomenon from the motion of the atmosphere and emerges in models built from equations that describe the atmosphere.