The reason Riot won't implement Solo Queue.

TL ;DR Riot made a smart business decision.

  A lot of people have been complaining about the lack of Solo Queue, and how Riot is extremely stupid and won't accept their mistake, but from a business standpoint, it makes perfect sense.

 League of Legends has been out for quite a while now, and to be blatantly honest, eventually everyone will quit this game. Everyone will eventually get bored of this game and quit playing it. But the people who will play it more, are the people who have friends in this game. Riot won't be able to please both Solo and Group players, so they cater to one group of players, Group players.

"But why not just implement Solo Queue as well?" Because that costs money, in order to run servers to keep Solo Queue up, they need to spend money on maintenance, and a bunch of shit so they can provide that to people. Eventually, these Solo players will quit, and re-implementing Solo Queue will most likely cost more money than it will make. Group players on the other hand, will play with their friends, they will play the game longer because their friends will keep them coming back, they'll enjoy the game longer, and will make more money for Riot. Solo players don't have too many friends to gift skins too, and for the most part, won't buy too many skins for themselves either. Group players like to show their skins off to their friends, Solo players don't give two shits about what their teammate thinks.

 As stated before, eventually every player will quit League, and Riot understands that. Catering to a more Group player focused game will keep Group players playing longer, and a lot of people are Group players.