Thanks for asking!
I think we would say that God did “build in” certain things: the laws of physics, the complexity and intelligibility of the universe, our ability to reason, etc.
By “His Law” I’m assuming you mean moral law. The Church does have the concept of Natural Law, which basically argues that through our reason, human societies will always tend to value the good and hold moral precepts against bad behavior.
Murder is bad, stealing is bad, share with others, hospitality, etc. These are things that don’t really need God in order for us to naturally conclude they are right or wrong (similar to Kantianism).
However, this only goes so far, as much of Christianity goes deeper. It states that love is dying to yourself for the sake of the other, turning the other cheek, loving your enemies, praying for those who persecute you, etc. These inclinations are generally not always found in nature (save maybe dying for others), and rely upon divine revelation for us to have learned them (or an incredibly insightful man I suppose).
Now, if we could conclusively prove that God exists through his signatures in the physical world, it would remove our free will, as we would not be free to reject God. If we are not free to reject God, it makes God a rapist who forces us into relationship with Him.
But because God is love, and an integral component of love is the freedom to choose that love; God can never completely reveal Himself to us without compromising our freedom, and thus our ability to love. So we are left to follow the breadcrumbs so to say. In this world, we will not encounter anything which will force us to believe that God must exist, as God desires the greater good of us freely choosing to love Him, rather than loving Him because “we may as well if He exists.”
As to your last point regarding Mary, and why God did not create (or recreate) everyone perfectly and give the world only those who would use their freedom for good… I have this answer:
Suffering in this world, caused by the freely chosen immoral acts of others, is a sort of crucible for the soul.
Encountering this suffering forces us to react to it. Will we respond by giving of ourselves, or respond by giving into evil?
I often think of Mother Teresa encountering the radical poverty of the slums. Had she not encountered this evil, would she have risen to the spiritual heights that she did? I don’t think so.
Further, if suffering did not exist, what is to say that Mother Teresa would even have left Albania? Maybe God would not even have given us the soul of Mother Teresa, because her soul, not being able to give of itself to others, would not find fulfillment, and her soul would fall into despair? In this despair she may have committed a sin, and thus God would not have created her free soul that would not sin in this hypothetical world.
Now let us imagine the soul of Hitler. He never encounters the horrors of WWI or Depression Era Germany, he doesn’t become radicalized, and he lives his life as an Augustus Gloop merrily drinking beer and eating sausages in Germany his whole life time.
In our hypothetical world, God would give us this soul of Hitler. Wouldn’t that be a sad thing? That the world receives the soul of Hitler because he wouldn’t use it for evil (because he never faced evil), and never would be graced by the soul of Mother Teresa (because she never faced evil)?
So I would reply that a world populated by free souls that would never sin is not as great as we imagine it would be. Rather it would be bland and full of people who are neither greatly good nor greatly evil. This lukewarmness is unattractive in the life of love.
God wants our full and whole selves, molded through the crucible of this suffering world. Anything less would be God creating a world of souls akin to paint by color rather than Rembrandt.