Gem mines should be available at town hall 9. They should produce 30 gems per hour with a max hold of 1000. It could cost 2000000 elixir to buy it. This would help players get upgrades quicker to want to stay in the game longer.
First of all I want to say thank you for sharing your thoughts.
You should think about the fact that gems are where Supercell earns its money from. So gem mines would affect the economy of Supercell negatively. You should know that we already have ways to earn free gems now and then : Gem boxes, objects (which appear every 8 hours), achievements and best clan of the season.