Sports can be great when competetion is transcended by fairness, as in the Olympic ideal. And, I think it serves basic human drives, be they reptilian, including achieving goals, gaining health, and even matching an opponant.
Sports, like everything else, has fallen prey to money, avarice, drugs, cheating, and so on, because it is worshipped or covetted, losing its essence, which is similar to that of art, to nourish the soul. Art, in general, however, tends to appeal less to the limbic brain and body, than to the neocortex or heart. What is art, (and what is sport), is a blurry question - there is a Ven diagram overlap.
Having lived with many artists, I know that some of them can be, or become, highly competitive and dissolute. I have known art-snobs who were vain, petty, angry people, moreso than would be justified in the real world outside of their community. And, there has been a lot of stuff called art which has triggered much discord, and sometimes hate bordering on the energy of a soccer riot. - https://en.wikipedia.org/wiki/Robert_Mapplethorpe
(Another confusing meld: Video games!)
I really do agree with you about the negative effects of sports, for some reasons mentioned above. I also think a major problem is that spectators empathise vicariously and thus, (like experiencing stress in the office), they often do not have a physical or real release for their ramped-up energy. The negativity can internalise and even encourage illness or violence. I had a UK minister LJ friend who enthusiastically watched a big game on TV, and later wrote of a serious fight he had with his wife, never connecting the two. And, of course, alcohol was involved.
I get bored watching sports on TV, and long lost the fantasy of liking them, (goaded to do so by society). Other people seem naturally interested. Some people are negatively affected. But I do believe it is possible to watch sports without the negative effects, especially in the 'Olympian' way, as many people otherwise uninterested will come over to watch on very special occasions. (Paired up with professional watchers might not be a good idea, lol).
To some degree, it can be external factors which turn the sports viewer into an angry Hun. These may be social, psychological, economic or environmental. Meat. Beer. Kids. Watching and feeling all-powerful, whilst being relatively powerless in reality, can be a big factor. But, teasing out all of these factors would require a whole lot of reseach, which any sports-craving population would refuse to fund.