...a question worth taking the time to consider! :)
So, though God always existed in perfect community, because His very nature is defined by His perfect & extravagant love He desired to create us, in His image, to be the object of His affection in order that we may know Him like He knows us!
I would say, therefore, that His desire is twofold: first, that in knowing Him we would love Him with all our heart, mind, soul and strength, and that, 2) we would, in essence, become an extension of Him - His perfect community, His perfect love - by loving those around us w/the same love He has for us.
In other words, the more we know Him and are known by Him the more we will desire to make Him known in the world and to those around us as we reflect God's image back into this world that He created. Or to put it another way, the more we get how much God loves us, the more we are compelled to love and care for one another and for all of His creation.
So really, Jesus' response to one of the teachers of the law sums up all my ramblings when He said to "love the Lord your God with all your heart, mind, soul and strength and love your neighbor as your self."
I think the two keywords in all of this are "love" and "community."
Just look at the Garden of Eden and God's intent to walk amongst his creation. His desire is to dwell with us - thus the tabernacle, later on the temple, and then Jesus himself comes and is referred to as Immanuel - "God with us!" And that same Jesus is now the "head" leading the rest of His "body," the Church.
And why? Because God is a God who loves to create good things; who, at the very core of His being, is defined by sacrificial love - a love expressed by extravagant self-giving. And so, out of sheer delight, He creates a world and a people - in order to share his perfect community with us and to pour out His extravagant love on us!😊