The problem with such sayings is that they miss the most important point. God saves sinners. If the person is not convicted that they are a sinner (not just that they sin sometimes), they will not see their need of forgiveness for who they are. "Asking Jesus into your heart" is a recipe for false conversions. Lord Jesus will not enter an unclean heart. Potential believers need to realise that they are lost and separated from God by sin. They need to realise that nothing they do can change that. "God be merciful to me, a sinner" is still the cry that God recognises. And when that person cries out, we can show them God's way of forgiveness through Christ.
Yes, knowing you are a sinner headed for hell and can do nothing to save yourself. Then believe on Jesus Christ..that He is God, that He lived, died and rose again, and asking for forgiveness.