The human brain has 80 billion neurons and a 100 trillion synapses. I think you're underselling the processing power of that warm chunk of meat.
The real message of the last 15 years has actually been the opposite: if you throw enough processing power at it, intelligence emerges.
Moreover we've known for quite a while now that glial cells also participate in cognition and moderate learning (e.g.: [1]). When you take those connections into account the numbers get really staggering. 85 billion glial cells with trillions of protein channels facilitating communication between the glial syncytium [2].
[1] https://www.sciencedirect.com/science/article/pii/S193459091... [2] https://pmc.ncbi.nlm.nih.gov/articles/PMC5063692/
The real question is not how many "weights" the human brain has (neurons+synapses may or may not translate into "weights", and brain might be also inefficient for what it is), but rather how much evolutionary and social "compute" was necessary to pack everything into that capacity.