I agree with the OP, except rather than trying the make the world a better place, we're seeking to get as many people rescued as possible. I view the world as kinda like the Titanic. The ship's fate was sealed on its maiden voyage when it hit the iceberg: when Adam ate the forbidden fruit. At first the ship was fairly level. Now it's up-end, and sinking fast.
But other than that, yes. we are to warn everyone in love, pointing them to the Lifeboat (Jesus) with humility, gentleness, and urgency. I am not qualified to judge, but soon it will be time for the One who is.
I like this analogy..this world is like a sinking ship, all we can do is rescue those willing to be saved. Back to the same sex topic, I had a problem with this too BC I didn't hate the sin of homosexuality... well since ..someone really close to me is. But my pastor says we're to love the sinner but hate the sin(what they do)