>> On the high level, there is no "chess AI", "go AI", "image classification AI" and "dexterous manipulation AI".
As another poster said these are all tasks performed by different systems. For chess and Go AI it's Deep Reinforcement Learning with Monte Carlo Tree Search. For image recognition it's Convolutional Neural Networks. Importantly, these systems are very task-specific. You won't find anyone trying to beat humans at games using CNNs, for example, or using Deep-RL to do text recognition. Far from "a few creative tricks" these are systems that are fundamentally different and are not known to generalise outside their very limited domains. They're one-trick ponies.
The OpenAI paper on "dexterous manipulation" reported learning to manipulate one cube, the same cube, always, after spending a considerable amount of resources on the task. It was a disappointing result that really shouldn't be groupwed with CNNs and Deep-RL for game playing. The level of achievement does not compare well.
>> Anytime a next task is solved, there is a crowd saying it's not a "real AI" and that scientists are solving "toy problems".
This used to be the case a decade or more ago. In the last few years the opposite is true. The press is certainly very eager to report every big success of "AI"- by which of course is meant deep learning.
>> 6 years ago we were able to solve some Atari games from pixels. Today, that feels like a trivial exercise compared to modern techniques
6 years ago DeepMind showed superhuman performance in seven Atari games with Deep-RL (DeepQN in particular): Beam Rider, Breakout, Enduro, Pong, Q*bert, Seaquest and Space Invaders. Since then more Atari games have been "beaten" in the same sense, but many still remain. I'm afraid I can't find references to this but I've seen slides from DeepMind people a few times and there is always a curve with a few games at the top and most games at the bottom, below human performance. There are some games that are notorious for being very difficult to solve with Deep-RL, like Montezuma's Revenge which was claimed to be solved by Uber a couple of years ago however this was done using imitation learning, which means watching a human play. The result is nothing like the result in Go, which remains the crowning achievement of Deep-RL (and its best buddy, MCTS).
Bottom line: Atari games remain anything but a trivial exercise.
And the architectuers that play Atari do not perform as well in Go or chess, say. You are mistaken that it's simple to train the same system to do all of those things. The AlphaZero system that played Go, chess and Shoggi well enough to beat its predecessor (you will excuse me that I don't remember which incarnation of Alpha-x it was) had an architeture fine-tuned to a chessboard and pieces with discrete moves, so it would not be possible to reuse it to play Starcraft, say, or even tic-tac-toe. The cost to train AlphaZero is also very high, in the hundreds of thousands of dollars.