Not only are we a witness to the power of the gospel when we live out the Golden Rule, but we also promote peace and societal flourishing.
The Word of God shapes both how we wish to be treated and how we would treat others. The Golden Rule both engages the moral imagination in specific circumstances and is grounded firmly in the Word of God.
Considering the speck and log illustration, notice that Jesus does not finish his example by telling us that we are just sinners and to not help one another change. The way forward isn't simply picking each other apart, it's by being a brother, a loving member of the family.
In seeking righteousness through following a list of rules or focusing on our performance, we may feel superior or clean for a moment but that doesn’t make us a Christian any more than wearing a Joe Montana jersey makes us a good football player.