Hey, I can read Wikipedia, too.
Sport is all forms of physical activity which, through casual or organised participation, aim to use, maintain or improve physical fitness and provide entertainment to participants.
It then defines physical activity (exercise) as:
Physical exercise is any bodily activity that enhances or maintains physical fitness and overall health and wellness.
With those in mind, do you think a doctor would ever tell a patient that he/she needs to get in shape and then suggest motor racing as a way to achieve it?
Further, the etymology of the word, to which you pointed, is ambiguous at best when taken as a whole as defined by Wikipedia.
"Sport" comes from the Old French desport meaning "leisure". American English uses the term "sports" to refer to this general type of recreational activity, whereas other regional dialects use the singular "sport". The French word for sport is based on the Persian word bord, meaning "winning" or "win". The Chinese term for sport, tiyu [...] connotes physical training. The modern Greek term for sport is Αθλητισμός (athlitismos), directly cognate with the English terms "athlete" and "athleticism".
The oldest definition of sport in English (1300) is of anything humans find amusing or entertaining. Other meanings include gambling and events staged for the purpose of gambling; hunting; and games and diversions, including ones that require exercise. Roget's defines the noun sport as an "activity engaged in for relaxation and amusement" with synonyms including diversion and recreation.
I think it's unfair to base any conclusions about a word's modern meaning when its etymology is that disparate. Otherwise we'd have to logically conclude that motor racing is sometimes a sport in England, might be a sport in France, is a sport for one in Iran and is never a sport in China and Greece.