Re: Shall we play a game?
The point I think you've missed is that no-one "provide[d] it the logic about how to trim the decision tree". It figured out the board evaluation and move generation functions based on completely general (e.g. non game-specific) neural network learning algorithms using historical data and reinforcement from playing itself. And did so well enough to beat the world champion on the first try.
With a branching factor the size of Go's, that tree-trimming logic is the part that really requires "intelligence" - or at least that part of intelligence akin to intuition, which is historically something people think that computers lack, and why Go has been a target of the AI community for so long. Humans have been poor at explaining their intuitions about Go as heuristic rules (beyond a few basic good/bad patterns), so the "traditional" approach of hard-coding rules has resulted in computers that suck, because they search the wrong part of the vast search space.
With chess playing computers, there was a much easier argument that there was no real intelligence at play, because the hard-coded heuristic rules and the search algorithm were constructed entirely by humans, with the computer just supplying brute force execution. With AlphaGo, there's definitely 'learned insight' present in the form of the trained policy and evaluation networks in place of heuristic rules.
No, of course it's not a general AI - no-one's claiming that it is. But more and more of the domains that were thought to be difficult for even specialised AIs are falling - alongside this, witness advances recently in natural language processing, computer vision etc.. And if you look at our brains, they look an awful lot like a lot of these specialised AIs networked together. Read "Thinking Fast and Slow" for an interesting psychology-orientated take on how much of our thinking seems to be dominated by the type of intuitive functions that AlphaGo's just shown itself to be world-beatingly good at.
By the way, apparently DeepMind are currently working on a specialised AI that will learn to play any card game (rules, strategies and all) just by watching videos of humans playing... :-)