Neural nets have fundamental limitations, particularly their inability to generalise outside a narrow band around their training set [1].
For instance- you will not find anything in the neural network bibliography about learning such a fundamental concept as counting, in the sense of finding the number n+1 that follows from a given number n, for arbitrary n. A neural net can certainly be trained on that task for numbers in a range [k,n] - but, given numbers outside this range its performance would "fall off a cliff". That's because neural networks can't learn general rules such as "x > y → y = x + k" etc.
Add to this the fact that neural nets, while very good at specific tasks, must be trained anew for each new task- and you see that there is a big problem in getting anything approaching "general intelligence" (which might involve any number of wildly varying tasks) just with neural nets. For one thing- even if you could train a neural net model for something like a million tasks, there would remain the question of somehow stitching them all together in a coherent whole capable of performing the right task at the right time or combining decisions from multiple models.
Finally, note that nobody is even trying to train a neural net to learn "general intelligence" end-to-end from examples; first of all, because we have no idea what constitutes general intelligence and therefore how to collect its examples.
So it's not a matter of scale. More of a matter that there isn't anyone living today that has any idea how to get to AGI with any existing technique- including neural nets.
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[1] https://blog.keras.io/the-limitations-of-deep-learning.html