Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
This is actually a very good question...
Agreed, if I would take a guess from interpreting several coinciding fathers opinions about tangent issues, I would say that the Father, Son , Holy Spirit exist as One by Nature, not will, generation, or procession. Generation and procession are I believe in the context of "how" less than "why"
or "what" and "will" I would say is definitely not the cause and further "will" is philosophically a "why" question. Surely Will is a part of it, as the Father must in some way Will the existence of the Trinity in order for the Trinity to exist, for all things including His self-existing Self exist according to the Will of God. However, we can't say the the Son and Holy Spirit ONLY exist by a willful act of the Father, because this negates their Co-Equality and further naturally subjugates the Son and Holy Spirit under the Father.
One key teaching about the Trinity is that we see an act of love within the Trinity.
So true, the Trinity reflects both the Love and relationship of God even within Himself.
2) The question, as posed, may well be unanswerable, it seems to presuppose that the Father could be the Father apart from the Son. But the Father and the Son are constituted by their mutual relations and are thus inseparable. How is it possible for us to conceive the possibility of the Father without his Son?
I think this answers the question from the OP