Since you are addressing Cold Hands specifically, and his absence, I'll float this (which is built from a comment I made on another, slightly overlapping question), which I think needs to be considered in its own right.
The reason Cold Hands isn't in the TV show is because...
Cold Hands isn't important
Before all his über-supporters jump on me, let me explain myself:
GRRM has been involved with the Game of Thrones TV show from its inception, and has obviously been advising the script department on the significance of certain elements, so they can draw particular attention to them. Much like one on the comments made about Harry Potter from @MattD above, this is to help with character motivation and overall focus.
If he was an important character to the Grand Narrative of Game of Thrones, wouldn't GRRM have insisted on his presence?
Think about it: he has had every opportunity to declare to the Showrunners that Cold Hands is the future king of Westeros, or has some significant part to play that couldn't be substituted by another character. If he was integral to its story, he would be included.
That isn't in any way to diminish his presence in the books, its just to say that he isn't required for the TV show. It functions perfectly without him. The only people who are concerned about his absence are people who perceive the show para-textually alongside the books: try it, ask anyone who hasn't read the books if they think its confusing or lacking without Cold Hands, and watch as they stare back at you with confusion.
Some fat has to be trimmed, and GRRM was obviously happy with them losing Cold Hands: he was in collusion with the show, and as such an accessory to his removal. He can't think much of the character, or at least doesn't consider him to be important...