The Holy Spirit does not convict the world of sin, the world hates God and loves sin. So how can the Holy Spirit be convicting the world of something it loves.
Unbelievers are spiritually dead and unable to choose God on their own. The Holy Spirit must first change the human heart and plant a new nature (being born again) so the person can see and believe the gospel. This inner quickening by the Spirit causes faith, meaning faith is the fruit of the new life, not its cause.
While the Spirit's initiating work comes first to make a person alive, the permanent indwelling of the Spirit is tied directly to union with Christ and faith. There is no stage where a person is permanently indwelt by the Holy Spirit yet remains an unbelieving, unrepentant outsider. The moment the Spirit regenerates, the response is immediate faith and repentance.
Quickening IS convicting!
God wants everyone to be convicted, but He enables humans to resist/ignore His will.
This truth is called God's intentional and permissive will.