Ultimately our faith grows because of God, but we do have our part to play in growing our Faith. Simply put, to grow in faith means to grow spiritually. It is to mature in both knowledge of God and in godly living; ultimately, it is to become more like Christ. The same way we grow from infant to mature adult, a Christian's life is designed to grow spiritually from baby to mature Christian.
"Like newborn infants, long for the pure spiritual milk, that by it you may grow up into salvation—if indeed you have tasted that the Lord is good." 1 Peter 2:2-3
Growing in faith requires both time and effort. God ultimately causes the growth, yet He expects us to follow His ways to grow, and become mature followers of Jesus.
Show Me Your Ways
I can agree with much of what you have stated. However,
1. There are primarily two sources for the origins of our faith. Not just God.
Faith would not exist apart from the power of God in organizing an environment that fosters the possibility of faith. (God)
Faith does not exist in us apart from another person passing their faith on to us. Whether it be family or whether it be those preaching the faith of the Gospel, mankind certainly passes their faith to others that are part of the human race.
Evidence for these facts are found throughout the Scriptures.
Rom 1:16 For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek.
Rom 1:17 For therein is the righteousness of God revealed from faith to faith: as it is written, The just shall live by faith.
The origins of Timothy's faith in his family
2Ti_1:5 When I call to remembrance the unfeigned faith that is in thee, which dwelt first in thy grandmother Lois, and thy mother Eunice; and I am persuaded that in thee also.
2. Faith grows because of our experience.
As you mentioned we must become little children in faith. Children are at the whim of their upbringing. They follow not only what they are taught but inherit the errors associated with their teachers. We are descendents of sinful people who both originate our faith but also "taint" our faith with their own. We become no greater than the ones who teach us. In fact, most of the time, we let others believe for us. Such has never fully satisfied God. Never.
Heb 12:10 For they verily for a few days chastened us after their own pleasure; but he for our profit, that we might be partakers of his holiness.
Me must willing admit that we are not what God wants us to be in faith. If such were true, this world would be different. What goes on around us is proof of our failures in faith.
Notice how experience "GROWS" our faith.
If we have the entirety of the very mind of God, then why the need for experiencing God?
Though we get our faith from others, it must ultimate become OUR FAITH to please God. We must agree with God of our own volition. We see God for who He is ourselves.
Job 42:5 I have heard of thee by the hearing of the ear: but now mine eye seeth thee.
Job 42:6 Wherefore I abhor myself, and repent in dust and ashes.
1Co 3:11 For other foundation can no man lay than that is laid, which is Jesus Christ.
1Co 3:12 Now if any man build upon this foundation gold, silver, precious stones, wood, hay, stubble;
1Co 3:13 Every man's work shall be made manifest: for the day shall declare it, because it shall be revealed by fire; and the fire shall try every man's work of what sort it is.
1Co 3:14 If any man's work abide which he hath built thereupon, he shall receive a reward.
1Co 3:15 If any man's work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire.
1Pe 4:12 Beloved, think it not strange concerning the fiery trial which is to try you, as though some strange thing happened unto you:
Our faith is endlessly dependent upon our own acceptance of God's ways.