Hmm a Reset Level (and Reset Story) would be cool for people who wish to do that.
Though you still wouldn't get a true new adventurer feel unless you reset or self-abstain from certain other things. Without using a new alt that means you'll have all your resources to acquire good gear early, something many new adventurers do not have (except for when the game hands it to you). You wouldn't have, as a low level, to worry about whether your chocobo will be useful (like when a new person gets one, puts dps skills, then realizes what they really need is the healer skills but don't have the ability/seals to buy the chocobo skill reset yet and get stuck with their choice).
So unless you're willing to reset your chocobo's skills to imitate that of a new player AND not use any gear (other than for glamour) AND purposefully put away all your gil so that you can start from fresh and act like you're brand new (and spend all your tomes/seals/other currencies), you will not truly get that new adventurer feel.