One God in three presences

Martin Luther once said, "to try to deny the Trinity is to endanger your salvation. To try to comprehend the Trinity is to endanger your sanity." The reformer chose not to risk losing his marbles but some of us Christians are less willing to simply accept such a central tenet of the faith as an incomprehensible mystery.
Over the years I have heard and read many attempts to explain the doctrine of the Trinity in a way that is both coherent and does not stray from the bounds of orthodox Christian theology. I have also spent many hours trying to figure out for myself how to make sense of the idea of three Divine Persons in one Divine Being. None of the explanations that I have heard, read, or attempted to devise, have ever stood up to rigorous scrutiny no matter how subtle, complex or seemingly clever they were or how sophisticated the mental gymnastics involved.
Ultimately, all attempts to explicate the doctrine of the Trinity come up against the following impasse: If each of the Persons is no less than God (and this must be the case, for orthodox theology requires that each is fully God, not partly God or part of God) and each of the Persons is no more than God (and they cannot be, for Christian theology teaches that each of the Persons has existed from the beginning, and in the beginning there was only God - nothing which was not God could yet exist, for God is the creator of all things) and if there is only one God then it follows logically from these premises that there can be absolutely no distinction between the Persons - each just is the other; in fact, there must really be only one Person. Yet Trinitarian theology teaches that each of the Persons is really distinct; though there is only one God, He exists eternally as three Persons, each of whom is not the other two.
It seems clear to me that there is no way around this conundrum and I have come to the conclusion (perhaps conclusion is the wrong word as my views on this subject are constantly under development) that since it is internally incoherent and logically impossible (not merely impossible in the sense of being beyond human experience – that, of course, would not be a problem where God is concerned - but impossible in the sense of being self contradictory; for a thing cannot be both x and not x simultaneously, neither can there be both three of something and only one of the same thing simultaneously, no matter how many mind-bending theories one posits about ‘three whos and one what’, relative identity or different senses of the word 'is') then it must be the case that not all of the tenets of the doctrine are literally true. In the words of Sir Arthur Conan Doyle, "once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth."
It seems, then, that the only reasonable understanding of the doctrine of the Trinity must be a symbolic, figurative, parabolic or poetic (as opposed to literal) one. In other words, it must be the case that either the Persons are not literally distinct from each other; or that none - or only one - of the Persons is literally God; or that God is not One in a literal (mathematical) sense.
Although I contend that the doctrine of the Trinity cannot be true in the literal sense, it is important to remember that just because something is not literally true does not mean that it is not actually true - it can still be 100 percent true but in a sense other than that implied by the straightforward, 'face value' meaning of the words.
My own understanding of the doctrine of the Trinity is that it expresses and symbolises the idea that, though there is only one God, Christians specifically encounter that God in three main ways; firstly, as the omnipotent, omnipresent creator of the universe and spiritual parent of all human beings; secondly, we encounter God in the life and teachings of Jesus Christ, who for all Christians is the ultimate revelation of God's character and disposition towards humanity; thirdly, we meet God in the life of the Church and in the lives of individual believers as they live out their faith, empowered by God's spiritual presence.
Another way of saying this is that God, our Divine Parent, is fully present in and revealed by Jesus (who is, in a special sense, God's Son) through the agency of God's Spirit - which is not merely a blind force but a personal Presence of love and compassion. This view of the Trinity could, perhaps, be described as one God in three presences (rather than Persons).
There are, of course, other ways one could understand the doctrine of the Trinity non-literally but, for me, this is the interpretation that makes the most sense in terms of reflecting both the structure of the Trinitarian formula (albeit in a figurative way) and my actual experience of the Christian faith.