In a post called "Sunsetting Clubs" League of Legends' team explained why the old club system would soon disappear in favor of a more modern solution.
The most obvious reason is the way players communicate nowadays. While some still use options like Teamspeak or Mumble, a big part of the gaming community moved to Discord. Using the vocal chat directly in League of Legends is less popular and the weight of the club system on the client is, therefore, less justified.
As explained by Riot Games, the technology used to build clubs is now kinda old and doesn't fit the client anymore. Improving the system would be too great of a risk as it could end up not meeting the players' expectations.
On the other hand for players eager to keep the community they built via the club system Riot games announced a new partnership with Discord before shutting down the club.
Original content by "Antagoniste".