Let me give you a straight answer... That belief has two things going so wrong with it. Let me explain.
First of all, we must understand that sin has tainted our whole being. That means that whatever we have to offer God is already tainted with sin. Actually, not just tainted. They're all yucky in God's sight. Isaiah 64:6 tells us that the best of righteousness we can give God still appears as "filthy rags" in His sight. Imagine how He sees our sins! Why is that so? Because the Bible is clear that God is perfectly holy! With this in mind, we should understand that man's sincerity does not, in any way, impress God. Behind the sincere facade we show God is a wicked heart whose inclination is rebellion against Him, and He sees that so clearly. Our sincerity is like giving God a well-decorated cake outside but inside are dead rats molded together as a cake sponge. Sin, which has so penetrated our being, is very despicable in God's sight.
A second reason why sincerity with religious beliefs won't do the work for our salvation is because the sincere person ignorantly misses the truth that the sincerity of working his way to Heaven is boasting in God's sight. God has made it clear that salvation is His work and not ours. The Bible is very straight in its declaration that we cannot earn salvation for ourselves. Only God can give that to us. Jesus, in John 14:6, said that He is the only way to the Father. Acts 4:12 declares that there is no other name under Heaven that can save but Jesus. Ephesians 2:8-9 makes it clear that salvation is a gift we receive through faith and not through good works which may lead to boasting. And so, by demanding that God looks at our sincerity as His basis for giving us eternal life, we are actually telling God that it should be our way and not His that should matter, and that is boasting! This is pride and James 4:6 tells us that God resists the proud but to the humble He gives grace. If we show God an understanding that there is no way we can save ourselves from the punishment of our sins and that He alone can grant salvation to us, the flood-gates of God's grace is opened to us.
It is out of the graciousness of God that salvation was made available. The divine standards of God makes it impossible for us to earn salvation for ourselves. And so, God, in His love and mercy, sent His Son, Jesus, to die on the cross so that the payment for sin, death (Romans 6:23), is made. We only need to come to Him in humble repentance and faith to receive the salvation He offers to every one who has sinned against Him.
This is salvation. This is grace.