Looking at your top-level answer, without wanting to be unduly argumentative, it's certainly reasonable to argue that the classics and the Bible have -- more than most other "knowledge" -- to date failed to save us from ourselves. They've been around thousands of years, and we still have massive poverty, greed, inequality, injustice, war, exploitation of natural resources at the expense of public health, etc. etc.
Are you then one of those Utopians who thinks that human nature is perfect, but "society" is corrupt and to blame for all the problems? On the contrary, it is human nature itself (at least since we were kicked out of Eden) which is fallen, greedy, unjust, etc. You cannot solve mankind's problems by burning the libraries. Instead, we must look into the libraries for wisdom that may help us.
For example, Federalist 51: It may be a reflection on human nature, that such devices should be necessary to control the abuses of government. But what is government itself, but the greatest of all reflections on human nature? If men were angels, no government would be necessary. If angels were to govern men, neither external nor internal controls on government would be necessary. In framing a government which is to be administered by men over men, the great difficulty lies in this: you must first enable the government to control the governed; and in the next place oblige it to control itself.
Setting aside your utopian strawman, if your argument is going to based on historical timelines starting with when "we were kicked out of Eden", I suspect we're not going to find much common ground on history, philosophy, or morals anyway.
The original question was, what should we leave behind to future. I think ultimately it's a great question, and a great debate. Let's agree to disagree on our approaches to it.
If (a) then you are a Utopian who believes that evil is to be blamed on history or society and not men themselves. If (b) then it seems you are a fatalist who believes that there is no hope for the hypothetical survivors anyway. I don't think (b) is the case because you now mention that you have an "approach" to the problem. It is clear that we disagree on our "approaches", but what is your approach? If you believe (a), then it seems the wisest approach would be to try to keep the surviving children untainted by words or thoughts from adults who remember the cataclysm, in hopes that the new generation will start from scratch. Am I wrong?
My point is that we don't know which part is useful and which part isn't, and to presume otherwise -- ie to believe that we can effectively, if you will, "play God" with some future society after we've blown this one up -- is hubris (not to mention likely contraindicated in the Bible somewhere).