An excellent post closed by a provocative (not negative) question. A few moments on Google brought up some possible replies.
" The fan idolises players, but often knows little about the club's history. When a club does not do well, more and more seats will be empty, whereas the section of the supporters is as full as ever. A supporter loves the club, not the team and its players."
Supporter definition: "a person who approves of and encourages a public figure, political party, policy, etc."
Maybe those who post only negative comments consider "support" as meaning disapproval and "encourage" as meaning negative criticism.
A little strange that "put downs" of club and players seems to be the new definition of giving support, although of course all are entitled to express an opinion (and that includes a positive one).