The character cannot turn out to be right. They are clearly wrong. But there can still be many opportunities for a happy ending. There are also many opportunities for twists, just not about the character being right. The character is wrong.
Yes, but does the player know this?
I mean, suppose I make a game where the character believes his friend is still alive despite seeing him die (and the player saw it too).
If you were to play a game where the protagonist wants to discover the truth and prove everyone that his friend is still alive, wouldn't you think "yeah, maybe in the end we discover that the friend faked his death"? Even if in the end it's clear that the friend actually died and the character was wrong all the time.
I am in total agreement with you on this. How can the player be completely sure? Even if the character sees a pig flying and everyone tells him that pigs can't fly - can the player be sure that the game is not taking place in a world where they can? Wouldn't it be natural for a player to think that the character is on to something? I do not believe that these thoughts and doubts can be avoided. The player wants to relate to the character. (in most cases at least)
"Wouldn't it be natural for a player to think that the character is on to something?"
Not if the game makes it crystal clear that the protagonist is wrong. What you have to do as an author, is come up with a story where going through the motions and letting the main character take you by the hand and following his deception, is really interesting.