While I agree that a horse not repsonsive to the bit exhibits a hole in its' training, if it is already hard in the mouth, just riding it from back to front may not change its' relationship to the bit and the meaning of the rein.
I personally don't think it's the end of the world to use the twisted wire bit for careful and limited retraining. When remouthing a hard horse, it is really a matter of applying pressure on the rein (one at atime to start with) getting the desired response and rewarding by stopping the pressure. It might have to be a lot of pressure to begin with, if the horse has no response to it because he doesn't remember what it means or he's been taught to ignore the bit long enough and the pressure will go away.
You have to not let the pressure go away until you get him to give. Then YOU give. you want to work toward being really giving, but you must always remember that the horse gives first. He EARNS the release. The more clear the relationship between him doing something and him earning a release, the quicker he will learn this and make it his normal way of going.
Your horse probably didnt' earn enough rewards so he saw no point in giving to the bit. And maybe he was ridden on endless, unchanging contact. He might hvae become defensive about it and try to pull the reins out of the riders hands. Is that what you mean when you say he is hard mouthed?
If you start with having him give to pressure and having him earn big releases he will see that rein contact is not a hopeless condition but that he can create release by giving. It's a contract between the two of you, and you had darn well remember to hold up your end; Don't give until he does and never forget to give when he does.