It's always the important jots and tittles one forgets. I'm sorry, that is should be an isn't.For one, Christians do not insist God is a logical contradiction. You might believe that the Trinity is a logical contradiction, but that is apart from it being the Christian's stated belief that God is a logical contradiction. Your objection here is completely irrelevant.
As to whether the Trinity is logical, I want to get at this from a particular perspective. When we analyze a doctrine like that of the Trinity, we seem to be presented with a plain logical contradiction; God is One and God is Three. We can always introduce certain distinctions, but this glosses over a fundamental point which the Trinity hinges on. That is whether or not God must be enumerable.
For something to be enumerable, it follows that it can be brought to a parity with meaning. That is, there is some proposition that captures its whole reality and sets forth a set of logical entailments that constitutes its own paradigm.
Can God be enumerated in this way? I think we have good reasons for believing that this cannot be the case.
For one, God is the foundation of everything. He is the reason that there can be anything at all, and what there can be and how it can be is dependent on His reality. He is the principle of being, the source of order and reality. This includes the possibility of something's being enumerated.
However, if enumeration is in this sense posterior to God's reality, then God cannot be wholly enumerated, because He is prior to the possibility of enumeration. In fact, He founds the possibility of enumeration.
What does this produce for our analysis of God? It doesn't seem to imply that we can't speak of God whatsoever. At most, only that we can't speak directly of God; instead, we pick out the pillars of reality and see where they logically resolve. We can lay down straight lines which pick out a tangent on the circle, as it were. So we can still speak sensibly of God, only with that perpetual qualification on our language.
As such, this leaves wiggle room for conclusions about God which don't seem resolvable to each other. It is only required that they don't be contrary.
Is it contrary to say that God is Three Persons? No. Each Person is God, yes, but this is because God is necessarily those Three Persons. Unity and Trinity can't be resolved by our language, but on the other hand they can't be shown contrary. This is a philosophical justification for the possibility of revelation which provides access to these divine realities not otherwise known by reason.
