The only way to make sense of the doctrine of the Trinity?

I regularly come across articles and posts on the internet which claim to offer coherent explanations of the doctrine of the Trinity yet, however promising they may seem at first glance, on close examination they almost always prove to have one or more fatal flaws. The majority of attempted explanations of the doctrine of the Trinity fall down in one of three ways. They either end up with no real distinction between the persons (the husband / brother / son analogy), or with distinctions that cannot exist simultaneously (eg. water / ice / steam) or – most commonly of all – with the Persons of the Trinity as parts of God rather than each of them actually being God as the doctrine requires. Many allegedly coherent explanations of the Trinity involve the idea of God as one entity with three ‘minds’ or ‘centres of consiousness’. The trouble with such models is that the only elements in them of which there are three instances (and therefore the only elements which are capable of representing the Persons) are the ‘minds’ themselves, so that on close analysis it becomes plain that the Persons are not actually depicted as God but simply as minds or centres of consciousness of God. And if each of the Persons is not merely a mind of God, but is the actual Being of God Himself, then, since there is only one Being of God, it follows that each of them is the exact same thing - in which case we end up with no distinction between the Persons.
Another of the most common ways in which the coherence of the doctrine of the Trinity is supposedly defended is to say that there is no contradiction between God’s oneness and His threeness, since the oneness applies to what God is and the threeness to who He is. God is, in other words, ‘three whos and one what’. But the trouble is, if each of the ‘whos’ is the ‘what’ (because each of the Persons is God) and there is only one ‘what’ then, once again, each of the ‘whos’ must be the exact same thing, of which there is only one. So, again, we end up with no distinction between the Persons.
I believe that grammar is the logical component of language – a universal element of all human languages, because logic itself is universal truth in the same way that mathematics is. The doctrine of the Trinity – at least in its traditional, orthodox form - is, it would seem, almost impossible to explain coherently in a way that does not defy the limits of grammar - and therefore the very bounds of logic and rational coherence.
There are, however, two ways in which I have found that it is possible to describe the Trinity without abandoning the normal rules of grammar. One is to view the word God as referring to a ‘mass noun’ rather than (or as well as) a ‘count noun’. Then we could say that while the Father is a Being called God, the Son and the Spirit are each God in the sense that they consist of a substance called ‘God’, even though neither of them is a God or the God (similarly to how a piece of an apple is still called 'apple' but not an apple or the apple - or how a piece of cake is called 'cake' but not a cake or the cake). However, I rejected this approach for the simple reason that this is not the way in which the vast majority of Christians (or others in the Abrahamic tradition) understand the meaning of the word ‘God’. He is not a substance in the material sense, or even analogous to a material substance.
The second way in which it is possible to make grammatical sense of the doctrine of the Trinity is by viewing the Persons in the same way we might view clones. For example, if a man called Tom made two clones of himself, called Dick and Harry, we could perhaps say that Tom, Dick and Harry, while being three distinct persons (because they are three distinct subjects (in the sense that only Tom can [truthfully] say “I am Tom”, only Dick can say “I am Dick” and only Harry can say “I am Harry”) are all actually the same man (three different versions of that one man, if you like). In the past, I have rejected this idea because, while it is possible to say that Tom, Dick and Harry are all the same man, it also seems perfectly reasonable to describe them as three identical men. Nevertheless, more than perhaps any other type of Trinitarian model or attempted explanation, this 'clone' analogy does, I think, provide a way to speak of God in trinitarian terms while not departing from grammatical (or logical) coherence. The Father is the one true God, and the Son and the Spirit are two other versions of that same God. One version of God became a man, the other two versions did not – God was crucified in one of His versions, but not in the other two, etc. Of course, the Son and the Spirit are not literally clones of the Father in the genetic sense - this is only an analogy, after all - and, unlike Dick and Harry, they had no beginning in time.
In my first post on this blog I quoted Sir Arthur Conan Doyle: "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." I have no intention of leaving the Christian faith (as the disciples said to Jesus in John 6:68 when he asked if they were going to leave him: “"Lord, to whom shall we go? You have the words of eternal life”) and, whether I like it or not, the Trinity is one of the fundamental doctrines of Christianity – about as near to a non-negotiable as it’s possible to get. So it’s important to me to have a way of understanding the doctrine that makes sense to me. This analogy of the clones, then, being the only way I can make sense of the teachings espoused in the Nicene and Athanasian creeds, is the model that I choose to adopt as my own understanding of the true meaning of the doctrine of the Trinity and as the closest we can come, in my opinion, to a true appreciation of the trinitarian nature of the Godhead.