This is correct. When theologians speak of God’s desires, they do not mean it in the same way as a human desires. God is already perfectly good and cannot change, thus he cannot desire to be something more. He is already complete in himself. (This is where the analogy in the OP is no longer accurate, as it paints a more anthropomorphic picture of God which is not literally true.)
Thus, God did not need to create the universe, even if he did do it. We cannot really explain why God created the universe, we can only affirm that he did. (In fact, that is probably the main reason why most of us think God exists at all: because we think the universe must have a necessarily existing Creator.)
Some Christians like to say that we are created “for God’s glory”, but that doesn’t mean that God created humans because he is lacking glory and needs to be worshipped. Rather, such a statement is referring to the intended goal or telos of humans, which is to imitate God in his perfect goodness as much as possible.
In Christian theology, we also affirm the existence of beings who are not physical, but are pure immaterial minds with wills: angels. It is traditionally believed that God created angels before humans and that they also fell into sin. So one could say that God did create both material and immaterial beings, but both of them fell into sin.
On the other hand, while humans are weak and fallible, being material creatures, it is also true that they can change for the better. Thus, God created Adam with the possibility of sin, but after the Resurrection those who are in Christ will be perfected into beings who no longer can sin. Why didn’t God create humans in this perfected condition from the first place? Perhaps because it is not possible to do that with material beings capable of free will. God has to go through the whole trouble of creating humans, seeing them fall into sin, and saving them because there really is no other way of creating material creatures capable of being in special fellowship with him.
One can see that for the most part, God does let the natural world “run on its own” according to the capacities and laws that he has endowed it with. In the few instances where he does intervene, it almost always to accomplish his plan of salvation for humans, such as in the Incarnation and the Resurrection. Why would God need to intervene for salvation? Because the goal of salvation is to accomplish something which natural things alone cannot accomplish, namely special communion and unity with God. As I explained on another thread, this supernatural bliss cannot be achieved by material means, which are necessarily limited. Even the most advanced civilization will ultimately succumb to the heat death of the universe.
Incidentally, the above is also a reason why many are dubious of creationists invoking a large number of miracles to explain the creation of certain natural things. If it is possible for a thing to be created via natural instead of supernatural processes, then it is more fitting for God to use the former, except if he has another goal in mind (e.g. the salvation of humans).