How to manage a mute PC?

I am GM in a custom universe I created (medieval fantastic), and a player asked me if he could create a mute character. I refused because I was afraid it would be too hard to manage, and not fun for him to play. Not to mention that the system I use is very basic and does not handle such a case, so I cannot rely on it.

By mute I mean a character who cannot make sounds with his mouth/throat. There is no universal sign language in this universe.

More specific concerns:

  • I am afraid the player will not play as much as before. He is a good player with good experience, but he sometimes thinks that RP restrictions forbid him from acting in some situations (if he is not the most skilled person in the group in a specific issue, he will not participate in it). How could I encourage roleplaying of the mute character (just like you would encourage a warrior to do things by putting battles in the story)?

  • As a GM, creating a place in the story for that kind of character seems almost impossible. He won’t be able to discuss his feelings or ideas with the other PCs and NPCs. As the stories I create are generally based on ethical discussions, information gathering from other characters, and are generally socially focused, I don’t know how to give the mute character his own piece of the cake. How can I give the player challenges and rewards while he is unable to do anything that involves discussion in a mostly social story?

  • A character who cannot express its ideas in the group is a sad character. What means do I have as a GM, and does he have as a PC, to communicate with the group in a easy and direct way, to make it immersive yet enjoyable?

  • Finally, if possible, I would like to know if you have any experience with this. I would like to let the player try, but I don’t want to ruin a whole campaign because of that decision.

EDIT : Regarding the answers

I accepted the answer because it displays the most practical advice, and experience, while covering all points of my question very carefully. The other answers here are still very good, and I suggest you look at all of them is you are in the same position as me.