I totally understand the incentives to not release the demo during the kickstarter. While a good demo would be a strong boost to the kickstarter a poor one would damage it. That said, if you wish to release a demo to the press I would advise agaisnt using the undead old one, you removed it from the game as you felt it was unrepresentative of the game experience you wish to produce, supplying it to the press with it would naturally give them a false impression of the quality of the game.
That said I am in favour of a one old one demo, possibly one that has already played through. (it depends, does starting on a scenario that has already been played through require the player to know more or less information than just starting one yourself? I suspect that a scenario that hasn't been altered will at least give the player all the information they need over time, while a scenario half played may have vital information about past actions permanently hidden from the player) If you cannot display a demo that isn't representative of the game you want to make, I'm sad to say the best option is to forego it for now. However, polish may not be that big an issue I'm not sure about the opinion of lets players, but I suspect most people who would back a kickstarter (or up their pledge) aren't looking for a smooth game from a demo this early on, they want to experience what they can do and how to do it, and gauge whether the experience might be fun.