You're sort of describing the problem of "over fitting", which is now very well understood in machine learning circles. That's when you get a model that describes it's training data very well, but doesn't generalise well.
The thing about using lots of data is that prior to publication of "The Unreasonable Effectiveness of Data" in 2009, most people did think that good algorithms were the most important thing. What that research showed was that a bad algorithm given more data will eventually outperform "better" algorithms, at least when those algorithms are initially judged based on their performance on smaller datasets.
So what happened with neural nets was that after some initial excitement about how they were more like the human brain, etc., it was found that "stupider" algorithms actually performed better and ANNs were written off for a while. It turns out that the reason NNs were performing badly was that they weren't being fed enough data.
Nowadays it's pretty easy to saturate a feed-forward neural network with data to the point where it's performance will never get much better. Deep learning techniques allow you to train bigger and more complex models with more data, but these more complex neural nets won't perform very well unless you feed them tons of data.
So in reference to your point about the brain, the thing about brains is that they actually learn based on massive amounts of data too. Think about how much data you have from continually streaming video ~16 hours/day, plus sound, plus touch, proprioception, and other inputs, over the course of many years.
Deep learning tries to emulate this to some degree with "pre training" which is where you feed lots of data into a deep network and have it learn "something" (it learns by itself at this stage). Then you start teaching it more complicated, high-level concepts. This pre-training allows it to do things like recognise common patterns in images, which the later training allows it to then associate with semantic ideas like "this is an apple", "this is a person", etc.
TL;DR: What seems to work best is fairly "dumb" algorithms, scaled up to be able to handle vast amounts of information and fed a ton of data to learn from. This is also how the human brain works.