Why do I have stable internet connection with everything EXCEPT league?

Because the internet as we understand it exists on a variety of servers in a variety of places, and your pathing routes to those places may not be the same as your pathing route to Riot's servers.

Which means that there could very easily be a bottle-neck that you only run into when connection to Riot, because that's the only server you try to access that routes through that particular bottle-neck.