One may, of course, argue that Go is a very sophisticated game. While a 30-node supercomputer could already beat a world chess champion in 1996, supercomputers with hundreds of cores were unable to beat a Go Master without handicaps until 2012.
However, your question is invalid, not because Go is a highly intellectual game, but because "the Four Arts of the Chinese Scholar" is a mistranslation of the term 四藝 ("four arts").
The Chinese character 藝 can refer to various skills, arts or crafts. E.g. 武藝 means "martial arts" and 工藝 means "crafts". Those arts or skills that are more intellectual and do not require much labour or physical strength fall inside the category of 文藝 (literally "literary arts", but it includes other intellectual or art activities/skills than writing novels or poetries). Some people nowadays translate the word 文藝 to "scholarly arts", but this is a poor translation if not wrong. Twisting "scholarly arts" into "arts of scholars" is a further distortion.
Even if we put the translation issues aside, there is no such thing as "the Four Arts of the Chinese Scholar" in classical Chinese. The phrase "琴棋書畫" did appear in ancient Chinese literature, but it was not meant to be representative of "scholarly arts". The earliest known appearance of the phrase is in the calligraphy commentaries 法書要錄 (circa 780 AD) written by 張彥遠. The relevant sentence says that
"Biàncái was erudite and skillful in various 文藝; he knew the very essence of musical performance, chess, calligraphy and painting."
(Remark: the 辯才 here is not the word for "eloquence" but the name of a person).
So, the author merely wrote that Biàncái was good at many 文藝 and they happened to be 琴, 棋, 書 and 畫. He didn't claim or imply that 琴棋書畫 were the four most representative 文藝 activities in Biàncái's or his own times.
In the prose 閒情偶寄 (1671), the author 李漁 did call 琴棋書畫 "the four arts". He wrote:
Those whose viewed themselves as ladies must know four arts, namely, calligraphy, painting, musical instruments and chess.
Yet, these are at best "the four arts of ladies", not "the four arts of Chinese scholars".
At any rate, Go is clearly an intellectual activity that should be classified as 文藝 instead of 武藝 or 工藝.