Why do we seek to be independent of asking anyone for help?
Because of pride in our life that is perceived from the world.
Why do we seek to be independent of God?
I think it is because of our soul. Our soul is usually in fight against the idea of being dependent on God. For us to be dependant on God the habit of relying upon the Father -- for everything should sank in our soul.
Why should we ask God to "give" us daily bread so long as we can earn a living for ourselves?
Since it is God who gives us the ability to earn a living, then in a real sense, it is he who "gives" us our daily bread. He strengthens us, and provides through us. So often, when we have our health, we take this ability for granted. Jesus is teaching us to look to the Father for every provision.