The quotation is completely wrong and should not be on the AGA web site without a disclaimer.
Leave aside the question of whether Confucius ever said anything that's in in the Analects - the sayings were allegedly compiled by his students after his death. Refer to the works of the likes of van Nordern if you want a modern view of who said what.
Leave aside also the question as to whether go was even mentioned at all. We can reasonably assume a board game of skill existed in the time of Mencius (372~289 BC), a fourth-generation follower of Confucius, because he refers to a master player Qiu. There is also a pre-Confucian reference to a game of skill in the Zuo Zhuan. But whether this game was go as we know it, Jim. is still open to doubt.
The reference attributed to Confucius is to the phrase 博弈. We are fairly sure 博 (bo) is a game of chance, more commonly known as liubo. The nearest modern equivalent may be ludo or pachisi, although liubo was possibly more advanced: it was hugely popular and acquired quite an extensive technical vocabulary. It may, for example, have been a dice game of skill in the sense that backgammon is. The primitive character form of 弈 (hands putting something on a board?) indicates the meaning may simply have been 'board game', and there is also a view that 博弈 is a binome just to pluralise the phrase into 'board games' (or, in the dour mindset of the western missionaries who first translated the Chinese classics, 'gambling games'). Binomes are rare in ancient Chinese (though not unknown). On that basis, other western translators preferred something like 'liubo and chess'. Ancient Chinese seem mostly to take the coordinate interpretation, but that is not quite the same thing as assuming 弈 is go. That connection came much, much later, in the Han, when Yang Xiong said in his book on dialects, Fa Yan: "Weiqi is [also] called yi. Eastwards of the Hangu Pass, throughout Qi and Lu, everyone calls it yi." (And that eastern area was also where Confucius lived.) But even in much more modern times 弈 has been taken to mean board games in general, and has been used, for example, for Chinese chess.
Going back to the Analects, the quotation refers to Book 17 (the Yang Huo book), Section 22. Yang Huo was a political schemer who wanted to employ Confucius (because of his prestige), but Confucius evidently disapproved of him and avoided him. Yang used a trick. He sent a pig (i.e. food) to Confucius's house when he was not at home. Etiquette required going to the sender's house in such a case, to express thanks. Confucius used the same trick and went when Yang was not at home. But eventually, it seems, they did meet talk, and if the conversations do reflect reality, we can see that Confucius did indeed regard Yang and others like him with disapproval. On that basis, we can infer that the reference to playing board games (if true) is to be taken as contumely. Confucius begins by saying that man who stuffs himself full of food all day (a possible reference to Yang Huo and the pig?) finds it difficult to put his mind to good use. He then allegedly adds: "Are there not players of 博弈? [At least] doing that would seem more admirable."
The character for the word I render as admirable was 贤 (which has overtones of 'morally good'). Legge (one of those dour Scots missionaries) rendered it, however, as "better than being idle." Later writers who 'translated' Legge rather than the actual Chinese have twisted this into weird things like 'trivial games'.
The 'morally inferior' sense of the original is backed up by the Book of Mencius where Mencius lists Five Unfilial Things (Meng Zi, Li Lou, [=Book IV, Part B, Chapter 30]). The second is "not attending to the nourishment of one's parents through playing 博弈 [boyi], and being fond of wine."
The first, incidentally was "not attending to the nourishment of one's parents through the laziness of one's four limbs", so Legge's reference to indolence was not as inappropriate as may seem at first sight - and even less so when you consider the old Scottish view of laziness!
The association of playing go and having a "good time" in China rings true. The riotous go parties on the Yangzhou lake boats are a famous example, and gambling was always part of go. But even in more modern times it was not quite the same serious obsession that we associate with the Edo players in Japan, where valorous tales of expiring att he go board or risking exile to dispute a promotion are the sort of things even western players have been brought up on. Segoe Kensaku remarked almost with horror on the way games were conducted by Chinese professionals at the "court" of warlord Duan Qirui in the early 20th century. Bystanders freely gave advice, and arguments would erupt after games along the lines of "if I hadn't taken you advice I would have won."
This all sounds like the early days of British go, where a loss by 2 pints was esteemed higher than a victory by 2 points.
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