I'm sure the Wardens studied every possibility for a cure, and have been since they first discovered the Calling. Unless you think the HoF is the only person with any sense of self preservation, or is just smarter than every other Warden who has ever lived put together.
Morrigan certainly didn't sacrifice herself for the world. She wanted the power. She always has. Saving the world might have been a side result, but that wasn't her motivation. The Warden has no way of knowing how accurate Morrigans information was. And even if they did think it was bad information, its not like they had any other leads to pursue instead. It's investigate what is probably bad information, or do nothing.
Once again, the well has no effect. Unless you are claiming game states that had Morrogan drink will have the cure, but not other states. The well has nothing to do with the Blight anyway. There is nothing to suggest it has any information about it outside of your own head. It could do, bit you could equally claim Morrigan just found the information in the Fade, or in the Skyhold library, etc. Nothing to suggest any of this in the actual game.
Finding a cure is important enough and big enough that's its going to be mentioned. It has not been mentioned.
Smarter than every other Warden, yes. Mine is having Avernus research the taint ethically, and in DA2 a breakthrough on the subject had been shown to have been discovered. It seems the Origins Warden and his allies are the only ones researching the taint and making discoveries.
The Hero had Avernus as an alternate avenue of research. Avernus had discovered a way to expand a Warden's life, which means the cure information was even better than what Avernus had found. I'm sure he consulted with Avernus, Irving, and others before leaving. Whatever Morrigan's motivations, she put her life on the line for the chance at helping the world however she could. She didn't do it out of desire to become more powerful. She wanted to help. So she wouldn't have given the Hero that information had she not been 100% sure that it was helpful.
If it had been weak information, then my Hero would have just stayed in Denerim and ruled over Ferelden along with his beautiful wife. The fact that this mission takes years is proof that there is something to it.
No, I'm claiming all worldstates will have the cure be found. The Well's knowledge being in Morrigan's mind makes the quest more likely to succeed, and thus all Heroes will succeed regardless. Her increased knowledge of the arcane and history will aid in the quest. And since BioWare likes to funnel wordstates into one outcome as much as possible, it will be the case here as well. The information Morrigan found is suggested to be linked to the elven histories. The Well holds more of that history, and Morrigan tells the Inquisitor that her elven writings make more sense to her now that she has the knowledge of the Well. So again, her newfound knowledge will aid in the search for the cure.
It hasn't been mentioned because it is big. They are saving it for the next game. They hint at it based on the rumors of a civil war within the ranks of the Grey Wardens.