Or maybe you don't let some convoluted system ruin your game, when they fail the "challenge".
Upon suspecting the Duke to be a demon, if there were clues being led to it, the party has already passed the challenge. Why then force them to backtrack to failure? Why not let them just go around asking and let them find the info needed to "prove it".
Why do we have skills at all?
Keep in mind, skills shouldn't replace RP, they should support it. Yes, the PCs go around town asking questions and searching for info. Their Streetwise checks (or whatever is appropriate) determine if they find stories that lead them to definite proof, or simply rumor and conjecture.
I'm honestly puzzled here. Are you suggesting that a skill like Gather Information or Streetwise shouldn't exist in the game? That if the PCs have some hints about something and want to learn more, they should always have it revealed to him?
If you don't want to RP it, then ak the DM for a summary of things found over searching, people met etc. You get the same result of the skill challenge which is contacts and info, without the needless chance of failure.
Why is the chance of failure "needless"? This is a dice based game, and part of the core philosophy is that not everything you do will automatically succeed. Even the way you describe this feels uncomfortable to me - rather than have the players dynamically explore the town and learn bits of knowledge, just handing them a summary of what there is to learn...
...I mean, I've done it myself, in situations where the info is relatively mundane or of little importance. ("Here are some basic facts about the town!") But it seems to otherwise be skipping past an opportunity for both RP and for some characters to demonstrate their strengths.
Possibly your example is just a VERY bad one, as it would leave failure as a chance that could completely screw the game up because of the mechanics.
-gather information (Endurance, Diplomacy, Streetwise)
failure here means the game is over
-find a way to break the curse (Arcana, History, Religion)
failure here means the game is over
-get to the Duke without being noticed (Athletics, Acrobatics, Stealth)
failure here means combat at least
Where are you getting "the game is over" from?
First scenario: The players have suspicions that the Duke is possessed, but can't find any proof. Do they start to doubt their own suspicions? Maybe they try to appeal to other authorities - or seek alternate ways to learn the truth. A sage might cast a divination for them if they perform a task. Or darker sorts might answer their questions for a price...
Or maybe they turn away from the area and investigate other possible suspects - until the situation in the Duchy grows worse, and they find themselves having to fight off a full demon invasion.
Second scenario: They've got proof but no way to cure the possession. Do they try and deal with it anyway? Sneaking into the castle to kill the Duke, even though that would cause great political chaos and leave them villified for their actions? Do they try and bring the proof to others, the chancellor or a high priest, and see if they can maybe imprison the Duke until a cure can be found? At the risk, of course, that instead the Duke finds out and they are on the run from his men?
Third scenario: They have the proof and the cure, but are caught breaking into the castle. Do they fight their way out, striking down innocent guards? Try to reason with them and present their proof? Surrender and demand a trial where they can reveal what they know? Have some PCs lead guards away so another can get to the Duke and perform the ritual alone?
Failures in skill challenges - in any obstacle in an RPG - should never* lead to a complete end of the game. They should just provide opportunities for new developments. Those developments might involve going into another challenge with fewer resources, or might involve having to deal with entirely new challenges.
(*Save perhaps in very rare situations - a TPK, or failing to stop some world-ending ritual, or the like.)
Nothing in the skill challenge rules says that failure should end the game, as you are suggesting. Indeed, the advice given regarding skill challenges often says the opposite.
Only in the most tightly-scripted plot-based games do you have a situation where something must happen in a specific fashion or the game ends - and honestly, in those situations, you shouldn't using a skill challenge anyway.