It's up for debate if NN's represent what is going on in our heads. But let's for a moment assume it is so.
Then indeed, an economist leverages a big set of data and assumptions about causal connections to speculate how this intervention would change the DGP (the modules in the causal model) and therefore how the result would change.
An AI could potentially do the same (if that is really what we humans do), but so far, we certainly lack the ability to program such a general AI. The reason is, in part, because we have difficulty creating causal AI models even for specialized problems. In that sense, humans are much more sophisticated right now.
It is important to note that such a hypothetical AI would create a theory, based on all sorts of data, analogies, prior research and so forth, just like economists do.
It does not really matter if a scientist, or an AI, does the theorizing. The distinction is between causal and non-causal analysis.
The value of formal theory is to lay down assumptions and tautological statements that leave no doubt about what the theory is.
If we see that the theory is wrong, because we disagree on the assumptions, this is actually very good and speaks for the theory. Lot's of social sciences is plagued by "general theories" that can never really shown to be false ex ante. And given that theories can never be empirically "proven", only validated in the statistical sense, this leads to a many parallel theories of doubtful value.
Take a gander into sociology if you want to see this in action.
Secondly, and this is very important, is that we learn from models. This is not often recognized. What we learn from writing down models is how mechanics or modules interact. These interactions, highly logical, are USUALLY much less doubtful than the prior assumptions.
For example, if price and revenues are equilibrium phenomena, we LEARN from the model that we CAN NOT estimate them with a standard regression model!
This is exactly what lead to causal analysis in this case, because earlier we would literally regress price on quantity or production on price etc. and be happy about it. But the results were often even in the entirely wrong direction!
Instead, looking at the theory, we understood the mechanical intricacies of the process we supposedly modeled, and saw that we estimated something completely different than what we interpreted.
Causal analysis, among other things, tackles this issue by asking "what it is really that we estimate here?".