Christians, have you done soul-searching after reading Luke 6:46?

The main change I recommend, and that I try to practice, is to give up the notion of proving myself right and everyone else wrong, because that eventually leads to worshiping myself instead of God and assuming God actually works for me. Rather, we ought to accept that we're probably wrong about something, and put our trust in God to sort it out and correct us as needed. 

I don't think the underlying attitude problem is exclusive to Christianity, or even exclusively connected to religious beliefs. People tend to pick sides and stir up divisions in politics as well. 

Sure thing - I'm mindful of that warning and have been for decades. Jesus utterly condemned hypocrites who claimed to be following him but who were not doing God's will. Read Matthew 7:21-23 about those doing mighty works in Jesus name, yet who will be disowned by Jesus as they weren't doing God's will. Yes, there's "much scope of improvement and correction in the practice of Christianity today." Always has been, always will be because Christianity is made up of imperfect sinners. Worse, many who claim to be Christians are not actually Christians at all. Their deeds prove that.

What change would I recommend? The Jesus of the Bible and the gospel of Christ being put to the forefront of all Christian preaching and teaching. A corrupted gospel is doing the rounds and that is the basic problem. People either think it's an easy-peasy-nice-an'-easy gospel or that they 'contribute' to it by doing all the right works/things (i.e. legalism). That is what stops people doing what Jesus actually says is to be done.

That verse is not about heaven and hell.  It is to the already saved person, that they ought to live right.  But if you apply it to heaven and hell, you will go to hell, because you can't add any works to salvation.  The Lord Jesus Christ is God, and He loves you.  Believing in the Lord Jesus Christ means being forgiven all sins past and future, and means going to heaven and not hell.  Death leads to immediate heaven or hell, and it is too late to be saved, after death.  All believers still sin.  See 1 John 1:8.  To be in heaven and not hell, believe in the Lord Jesus Christ, who is God and who died on the cross and shed His blood to pay for all of our sins in full, and who was buried, and who resurrected from the dead.  The only way to avoid hell is by believing in the Lord Jesus Christ, without adding any of your own works.  See Romans 4:5, 1 Corinthians 15:3-4, and John 3:16.

“Why, then, do you call me ‘Lord! Lord!’ but do not do the things I say?” (Luke 6:46) A Christian must conduct his own self-examination to prove to himself that he is following Christ. Others may say he is or is not, but unless he himself knows that he is, unless he himself sees his mistakes and takes steps to right his wrongs, all is lost.

The fruits of Christendom’s religious confusion are to be seen in its rising rate of crime, its juvenile delinquency and its alarming increase in divorce and godlessness. It is all what Jehovah foretold for ungodly Christendom in these “last days.” (Matt. 7:15-23; 2 Tim. 3:1-7 

Christendom denies Christ both in her doctrine and in her way of life

When I read your question this scenario came to mind. Is it familiar to you?

Luke 18:9-14

New King James Version

The Parable of the Pharisee and the Tax Collector

9 Also He spoke this parable to some who trusted in themselves that they were righteous, and despised others: 10 “Two men went up to the temple to pray, one a Pharisee and the other a tax collector. 11 The Pharisee stood and prayed thus with himself, ‘God, I thank You that I am not like other men—extortioners, unjust, adulterers, or even as this tax collector. 12 I fast twice a week; I give tithes of all that I possess.’ 

13 And the tax collector, standing afar off, would not so much as raise his eyes to heaven, but beat his breast, saying, ‘God, be merciful to me a sinner!’ 14 I tell you, this man went down to his house justified rather than the other; for everyone who exalts himself will be [a]humbled, and he who humbles himself will be exalted.”