The comment about cyborg chess is spot on.
This is why AI is more significant to Go than horses and cars are to running.
It's easy for humans to build machines that enable them to travel faster than they could run, but it has zero impact on running as a sport. The existence of cars doesn't change how runners run. But the existence of AI does change how Go players play.
Now that AI is more powerful than human players in chess, it is no longer possible to reach the top of human performance in chess without AI. That's why this is so significant. It's not that a human can't beat the AI. It's that a human can't beat another human without learning from the AI. You simply can't play chess at the highest levels anymore without playing cyborg chess. This is different from the impact of other machines on other sports.
But I disagree with you on the comparison to how we use the internet to enhance our other abilities, such as programming. Computers help us be better programmers, but they can't replace us. AI completely replaces humans as a Go player. The human is no longer needed at all.
I can see this causing a significantly different psychological response. In effect, a human trying to become the best chess player or best go player they can become is trying to more and more closely emulate a computer that they will never actually reach parity with. Another commenter noted that "A Professional Go player is an explorer of truth in a millenarian board", but this is less and less true as AI improves. All knowledge about the best way to play Go was previously gained by humans. In 20 years, humans will probably be meaningless on driving knowledge of perfect Go playing.
Chess players today rely on playing moves which their opponents don't expect and haven't trained on as much as they rely on outplaying their opponents. If there is a perfect chess game or a perfect go game, humans are not going to help find it anymore.
So AI doesn't make it that humans can't meaningfully compete in games such as chess and go, but it does change how they compete. We no longer drive the pursuit of perfect play as "explorers of truth". We no longer learn primarily from each other, we learn by imitating the more powerful, yet unmatchable, AIs. And we no longer compete primarily on sheer expertise in the game. We have to, in some sense, include trick plays that work by exploiting the humanity of our opponents rather than playing for a platonic ideal of the game and seeing who is better from that perspective.
AI has fundamentally and irreversibly altered these games, and I think it's appropriate in some sense to mourn the loss, regardless of our feelings on the new reality we live in.