Versäumnis is a failure when:
- You know there is a problem (poverty, lack of social well-being, crime)
- It is assumed you know how to fix it (when talking about world or society it often means: Adhering to the ideology of the speaker)
- The time to fix the problem has passed, at least for the characters relevant to the speaker.
In those cases it becomes a failure of omission.
Deine Mutter und ich waren ein Produkt der Versäumnisse unserer Welt
So to back-translate it to English:
Your mother and I were a product of the failure of our world to fix (in time) the problems we already knew we were having (as a society).
In other cases
Versäumnis is negligence, an oversight, or even a deliberate omission if resources are too constrained.
In German legalese
Versäumnis is also used if you are non-compliant with a ruling on time.
A related word is when you are defaulting on a credit rate, or you passed a work deadline:
säumig (defaulting / delayed),
säumiger Schuldner (defaulting debtor)
If you are too late for a movie you are
zu spät, but if it is over when you arrive at the cinema you have
versäumt the movie.
säumig words imply that the chance to fix something is over and you have to face the consequences.
Which is why it is a good choice by the translator to convey the meaning of the English scene.