Scripture teaches that worshipers, once purged, have no more conscience of sins (Hebrews 10:2). Of course, that's not something we see a lot in our experience.
One problem (perhaps even the main problem), is that we treat Christ as a sort of a safety net. As long as we're doing well – as long as we're walking more or less uprightly – we think that we're accepted in God's sight by virtue of our own uprightness. We only really think of ourselves as accepted in Christ when we realize we've failed to walk uprightly.
Of course we'd never say that, but our actions and our prayers reveal what's really going on in our hearts.
But God doesn't ever have a "Plan B." God has given Christ to us to be our righteousness, our holiness, and our redemption (1 Corinthians 1:30–31). It has never been a part of God's plan for us to be acceptable before Him other than in Christ. We are accepted in Christ (Ephesians 1:6) and only in Christ.
It's hard to remember that I am no more accepted before God when I'm walking well than when I'm walking badly: regardless of how I am actually doing, I am accepted in Christ. That's a comfort when I see myself fail, but it's humbling when I think I'm doing all right.
that I may be found in him, not having my righteousness, which [would be] on the principle of law, but that which is by faith of Christ, the righteousness which [is] of God through faith (Philippians 3:8–10)