This is heavily debatable but I genuinely believe that people do not play support because support is just not a rewarding role to play. In league an individual player can have a major impact on a game which majorly takes away from the team aspect of the game. Yes teams are necessary to win the game no player is 1v5ing his way to victory but this is not my point. Supports can make impactful plays in regards to the early game and with the team but alone playing a support (on equal stance with the enemy team) just does not feel rewarding to me. I do play support I mainly love thresh because of the CC and hook potential but when I queue up as a support I don't queue up thinking I want to play this role and look forward to enjoying it I think about how I am sacrificing myself so my team can enjoy themselves.
I am sure someone is going to go on a tirade about how I am wrong and supports are majorly impactful core members of the team and are completely necessary then they are going to downvote me because my opinion is somehow less than theirs but hear me out. When playing support (mainly thresh braum bard and blitzcrank keep in mind) I can not function as an alone player all of my abilities are centered on me helping my teammates. All of my teammates to an extent can function as a lone player but a support can not. A support needs someone to help because without that person the support is also helpless.
I don't want to go overboard with this explanation but in short I have played other games where playing support has felt a lot more rewarding but these games were much more team based and put no stress on an individual player besides the ability to function as a group.
The reason people don't want to support is it does not equal the fun experienced when playing one of the different roles in the game. The game puts too much stress on the individual player to perform well to make people want to take that role.