I've never experienced this. Maybe I don't play games with people who get to that point so I don't see it but if someone did reach the point of being a 6 year old about a game then I probably wouldn't ant them there in the first place.
Some people do and will act like a child when they don't get their way. I for one, do not entertain any kind of immature or moronic behavior in my space. If you want to play with the sand in your own kind of way regardless of others around you then go get your own sand box!
I would let them leave. It would not be good to play with someone who would always want things to go his/her way just to win. Maybe he could invent his own game where he could make his own rules and where he could bend them so he could win.
I only experienced this with one player, and to be honest, he only ever played Thief characters, and would always try to steal loot, or in some way gain some advantage over the other party members. (It reflected his real-life personality in a lot of ways).
I could tell the morale of the group was being severely hampered by his exploits, so as most of his cowardly acts were happening within the capital city of my campaign, his character soon drew the ire of some notorious npc's. This was the point at which the player threatened to leave the game, and accused me of being a biased Dungeon Master. The fact was that the local ruffians did run a cartel, and if he had half a brain he would have realised that it was part of the adventure, and that he had brought their attention upon himself. He had a legendary strop for a then 27 year old, and stormed out of my house. He tried to get the other players to join him, but they were keen to see him go, and his character was adopted by another player, who maintained the character's asshole-ness without damaging the party.
Even if the player has made financial contributions, you are the one who is putting in the time and effort to keep the game going. There are so may games out there that if their preferences are genuine they should be able to find what they are looking for, or even start their own. You will never be able to please everyone and should not even try. Just be polite, wish them well, and follow your own vision.