Games, like sports, can help you a lot, but they alone won't simply make you happy/non-depressed.

I could use a food analogy. Kale can improve your health, but if you're sick it won't cure you, it'll just have a positive impact in your overall health.

I'd like also to add Minecraft to that list. The ability to get in and just start building your world as you imagine them can be very relaxing.

Games with small communities are pretty cool as well. Once you're in, there's that belonging feeling that you don't usually get on popular games. Small games communities exist because the people there really love the game and their attitude to any newcomer is warm and friendly. Unreal World is a roguelike that comes to my mind writing this.