I totally agree with consciousness being an emergent property.

But can a bigger system have consciousness when individual parts of it are already conscious? I don't see why this should necessary follow. I'm sure there are somewhat similar and interesting emergent properties, but we'd probably call it something different. Probably closer to the kind of group identities we already have with lots of humans interacting. Though obviously we'll know what it feels like, just like none of my neurons will ever feel like i do.

I think a better way of putting panpsychism is to say: what you are doing with your science is describing how consciousness behaves. So when you describe a simple system with maths, you are describing the consciousness of that simple system. Describe a complex system with mathematics and you are describing the consciousness of a complex system. You don't need to describe the transition from simple consciousness to complex consciousness with anything other than mathematics because mathematics already describes the transition that arises from the composition of simple objects. What it feels like to be a simple object vs a complex object is outside of the scope of science.

What differs this scenario from individual neurons firing together in order to make the human mind? Can you keep taking that line of thinking down to the subatomic properties of the neurons themselves? I'm guessing this is the argument for panpsychism. The emergent property of human consciousness could be seen as the culmination of all the individual parts themselves have some form of primitive consciousness.

Isn't that argument roughly as useless as attributing the ability of a watch to show time down to each individual mechanical component? It's technically true, but not very useful to elucidate what actually makes the watch a watch. For example, it sounds kind of silly to say that a single watch gear has some form of primitive time-keeping ability, or has some form of primitive ability to tell me whether I'm running late. The universe is rife with examples of emergent complexity in which the whole is greater than the sum of its parts.

Well it is still useful but not to you directly. The distinction being that the right intermediary could make use of it.

If you had a physics model it could describe it perfectly but it would likely involve a very large group of matricies and linear algebra constraints. You technically could work through every equation given a large set but examining the watch yourself would likely be quicker, if not learning to be a sufficiently good watchmaker.

However if you assemble a computer model to handle every component's interaction individuallt and a physics engine to do number crunching you could get a working mathematical model of the clock.