I'm not convinced that the doctrine of the trinity is accurate.
That's a bold claim to make, I realise. Hundreds of years of development resulted in the familiar formulation that most in the western church have been taught. Augustine did a great job of putting it into words, to be sure, but here are my doubts. When I read Paul I see a distinction between the Father, the Son and the Spirit (and the spirit) more than I see a unity. He likes to write, "the God and Father of our Lord Jesus Christ" as a kind of distinctive title. It's at the start of most of his letters. The Holy Spirit appears as power which enacts God's will in the world. Lastly, "the spirit" is something different again, seen in the repeated use of "kata pneuma" (according to the spirit) in contrast with "kata sarka" (according to the flesh). By reading Paul alone I get the impression that he emphasises the distinctions and barely deals with the unity. That's not to say that the early believers didn't have ideas of divine unity; Paul also didn't pay a lot of attention to Jesus' miracles, signs, or sayings either.
On the other hand, the Johannine literature is significantly more interested in the unity of Father, Son and Holy Spirit. Jesus is one with the Father. Jesus wants the disciples to be one, just as the Father and the Son are one. The Holy Spirit prominantly gets a personal pronoun in the Johannine literature. The concept of the pre-existent Christ who is one with the Father is loud and proud. It's clear that there is a kind of relationship going on between the three, but it feels as though Augustine's doctrine of the trinity is always going to fall short of being able to explain it. The doctrine can't even be explained without first explaining Greek concepts of ousia, hypostasis, and prosopopeia. Such dependence on a single language gives me pause to doubt the accuracy.
The social trinity has a bit more appeal to me. Unity in the relationship gives us the opportunity to live out Jesus' prayer that his disciples may be one even as Jesus and God are one. It holds out the potential for the divine community to be more than a collection of individuals who have a common interest. The social trinity invites the participation of all people and allows us to transcend the self. Where I think it falls short a little is beyond the Johannine framework, or outside the Johannine texts. I think there might be the possibility of bridging the gap through Bonhoeffer's work in Act and Being and Sanctorum Communio, particularly that the being of the Christian is found in the being of Christ both as the self and as the other.
Perhaps the most awe-inspiring model of the trinity is the death of God model (for shorthand I like to call it the dialectic trinity). The pure kenosis of God, the complete and utter self-giving in the movement from transcendence to immanence astounds me. This is not the action of a God who wants to be in relationship with humans, but the action of a God who wants to be united with humans. The transformation of God and the transformation of humanity at one and the same time speaks volumes about the character of God. I find it counter-intuitive to pray to God in that way, but that could be simply a product of my formative years. I also find it to be an awkward fit with the teaching to pray to "Our Father in heaven" unless we are thoroughly committed to the notion that heaven is where God is, and since God is in the Church we therefore have heaven with us here and now. This is certainly not your patristic or classical understanding of God.
Just like the Augustinian model, both the social trinity and the dialectic trinity bump up against the limits of language. They're informative up to a point. They're deficient beyond that. I don't find it especially helpful to insist that it can only be understood through revelation either. Do we need a doctrine of the trinity? Well, yes and no. As informative as they are, they just aren't solid enough or necessary enough for Christian discipleship. I like them and I don't, hence my trinitarian ambivalence.