I would say that in the original sentence it only makes sense for ‘David’s’ to mean ‘David is…’
The entire sentence is:
David is arrogant and so self-righteous - you would think he had never done anything wrong in his life.
In addition, a native English speaker would not shorten ‘David was…’ in that context.
This person has done wrong things in his life and the speaker is fully aware of it. The expression indicates that the person should not be so arrogant because of the mistakes he has made in the past.