You have a trillion connection neural net wrapped in 2 pounds of flesh inside your head. This is a massively larger amount of hardware compared to just about every animal out there. Throwing hardware at a problem is a solution to intelligence.
I don't like the biological comparison, but using your metaphor it would be like God saying "Hey I've created a brain but only have 10 billion synapses. Evolution would normally take 10 years to get to human-scale at our current organic rate but if I throw money at building a bigger brain cavity I can squeeze in the 1 trillion to get there today!"
At the very least, it's interesting to see how much more accessible computation has become. Back when I was in school I could only dream of having a cluster of 280 GPUs. When sometimes the dream would come true and you had access to a cluster you would have to wait your turn in the job queue and hope you had enough compute in our resource quota to prevent your job from being terminated.
Now I could spin up a 280 GPU cluster on AWS (after dealing with pesky utilization limits) for only $182/hour. If researchers at Google have been doing this non stop for the past year they have "racked up" $1.6M on just compute. This is a drop in the bucket for a marketing department and the publicity they have achieved. I don't think normal Go AI developers have access to those resources :)
Modern chess engines are built on a testing infrastructure that makes it possible to measure how each potential change affects the playing strength. This "Testing revolution" has brought massive improvements in playing strength.
For AlphaGo, it's probably the training that requires the most computational resources. The 'distilled knowledge' could perhaps run on a desktop PC. The program would search fewer variations and would be weaker, but if AlphaGo improves further, that version might still be stronger than any human.
Also, it feels like training the AI against many games with lots of hardware is somewhat equivalent to a human progressional who engrossed themselves in the game and trained since childhood.