If all people who were striving to be apologists applied this aspect of Paul's ministry to their own, they'd be different and better people. Another way of thinking about this goes like this: we all know that non-Christians or ex-church people think poorly of Christianity because we are all hypocrites, mean-spirited judgmentalists, fundies in the worst sense, and people who care about money more than we do about anything else. There's no stunner or world-changing revelation in that confession from a non-believer.
And, most of the time, we sort of wear that like a badge of honor. I know I personally get a little bit of a kick out of people calling me a "mean Calvinist" -- mostly because after years of listening to people say that, somehow that's the last battlement for every argument which is falling apart.
But let's imagine something here: what if I approached every apologetic encounter with the personal conviction that I am actually a mean Calvinist. What if, rather than mouthing the words "chief of sinners," I approached apologetics from the standpoint that I was a man of unclean lips among a people of unclean lips, and since God didn't have an angel touch the hot coal to my lips yet, whatever I am doing I am doing it as a beggar among beggars, a vile man who is actually in desperate need of the thing which I say God is offering in Christ?
That doesn't mean I surrender truth or take an "I'm OK - you're OK" approach to people. It means that when I deal with them, I see both the sinful flesh and the image of God in them, and I appeal to the image of God rather than attack the rebel who has a hard heart.
See: I think that if we study Scripture long enough, and listen to what it teaches us about ourselves, real humility has to be among the convictions we receive. But to do that -- to become decent apologists who can approach people with the declaration that Jesus is Lord and Christ as Peter did at Pentecost as if it was important but not an insult -- we have to have a decent hermeneutic of Scripture.
This really does go back to the question of how we read the Bible. Because if we read it as if it is merely a handbook of apologetics, we're going to miss the other 81.7% of the Bible which is telling us not how we should view other people, but how we, who are called by Him name, should view ourselves -- how we should act, and think, and work, and love.