The Reason Support isn't Commonly Played...

...is because no one focuses the Support before the ADC. Wait what? Allow me to explain. In other similar games, excluding Dota which is it's own beast with no definied meta/ideal play style/etc., Supports are something that naturally define a team and are often focused since characters in their game often rely on Supports for specific mechanics or advantages that only a Support can provide. In Overwatch, you want to kill the Support first due to their insane utility and healing they provide. In World of Warcraft and Heroes of the Storm, you want to focus Support due to the amount of healing they provide a team. In TF2, you really want to kill the Medic/Engineer due to healing and utility they provide. You get the point. In League of Legends, no one feels the need to focus the Support because their team doesn't really gain anything unique or specific that would warrant focusing a Support. Due to this feeling that you don't need to focus the Support in LoL, people naturally see Support as the least impactful class and therefore don't find it interesting. The only Support that is focused is Soraka and she is usually banned when she is strong.

The point I'm trying to get across is, since the Support class is seen as the least important class to focus in a fight, they are seen as the least important/interesting role. Their big moments aren't seen as big enough to focus. No one will play Supports commonly until they are focused in team fights over the ADC.