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First of all, we must establish that the original text of Draconian constitution is lost to us. All we have are some fragments and anecdotal evidence of them being very severe as mentioned by Aristotle, Plutarch and others. What did remain of them in later Athenian laws (like Solonian constitution) are parts about killing specifically. In fact, as we can see, they were republished at a later date. Interestingly enough, they do not appear to be so harsh - punishment for killing without forethought was not death but exile, and the killer himself had certain protection under the law. About the person of Draco himself we also have very little knowledge. Except being Athenian and most likely noble, everything else is a matter of legends, even the manner of his death.
What we do know with some certainty is that Draco's laws were used as a sort of bogeyman by latter lawgivers. Something like, if you complain about harshness and inequality under the law, you would be reminded that our elders endured even harsher laws. The Solonian constitution is remembered as the one replacing Draco's laws, and although again we do have only fragments of it, it is supposed to be more lenient. Note that ancient writers like Aristotle use Draco as a symbol of severity, and do that as already established fact. This show that the tradition about Draco's draconian laws already existed a few hundred years after his death (Draco supposedly lived 650-600 BC, Aristotle 384–322 BC, Solon 630–560 BC). This also establishes that Draconian laws in their original form didn't last long.
This now leaves us with the question, why did Athenians even accepted such harsh laws? When we dig a little deeper, and of course if we accept that such laws actually existed outside of legends, we conclude that the severity of the law was aimed at lower classes. Supposed death penalty for stealing cabbage, but would a noble steal cabbage? Debtor forced into slavery, but only if of a lower status then creditor. On the other hand, the killing of a low status person could be deemed "accidental" and end in exile (even that could be later solved by "reconciliation" or pardon by ten members of the phratry ) . If Draco's laws ever existed, they were likely a codification of exiting practice with established inequality, rather then something new and revolutionary. As such, they only became draconian when Athenian society started moving towards democracy, and that did start in the time of Solon.