1. The understanding of doctrine is a major deficiency of modern Christianity. That is one reason why the Church is now dealing with heresies like modalism and unitarianism again. The modern church by and large is exceptionally weak in teaching orthodox doctrine. I say this to explain the cluelessness of the people you know about the physical body of Jesus being in heaven.
2. Confessing that something is a mystery is not the same as saying we know nothing about it. The Catholic Church, for example, calls all of Jesus’ life a mystery; “The Mysteries of Jesus’ Public Life” for example. Saying that the Trinity is a mystery is not simply throwing up our hands and adopting a “your guess is as good as mine” approach.
3. The Trinity is at the heart of all Christian theology and praxis. To be Christian is to be Trinitarian. You may be a lot of things, an Arian, a modalist, etc. and say you worship one god or Jesus, but you cannot be a Christian if you are not Trinitarian. These issues were important enough for the Orthodox and Rome to go separate ways, they are not small matters. As Vladimir Lossky has written:
For us the Trinity remains the ‘Deus absconditus’, the Holy of Holies of the divine existence, where no ’strange fire’ may be introduced.
4. When we bring the LDS Church into the discussion, the gulf is impassable. It seems to pour scorn on the Trinity, while trying to play at a ‘Godhead.’ Three gods, with the potential existence of more on Kolob, and an infinitely expanding number of gods in the future is not even close to the doctrine of the Trinity. If we delve into the details, I think we will find that at almost every point LDS theology is anti-Trinitarian and is tri-theistic or polytheistic. This is not simply ’explaining the mystery differently.’ This obvious and acknowledged anti-Trinitarianism is what causes me to be baffled when Mormons want to be thought of as Christians. Other than a PR move, why the desire to be thought of as Christian while denying the entire stream of Christian thought?
If I say that I am a Muslim, but believe that Allah is a woman, I can become another god, and that the Quran is not the final revelation of Allah, I may be a lot of things, but I am not a Muslim. There is a historical and doctrinal reality that is Islam, and I cannot redefine what that is and then say that my beliefs are Islam too.
As I said, Christianity is Trinitarian, period. The LDS church teaches against all kinds of Christian doctrines, and then says ‘why aren’t we accepted as Christians?’ If you want to teach all these other doctrines, fine. But don’t try to be historically revisionist and say that your new teaching is Christian. Why do you covet that label anyway if it is what Joseph Smith was protesting against?
The ‘values by which we live’ is precisely the point. Muslims may share those values, Hindus may share those values, atheists may share those values. Family, home, country, flag, sexual purity, honesty, etc. do not qualify as Christianity. They may be an element of it, but they are not the heart of the faith. The heart of the faith is the worship of the Holy Trinity, as revealed in His Word. Plenty of ‘good’ Muslims will die for their faith in Allah, and despise the Trinity, and Jesus as God. Yet they might live outstanding ethical lives. Do they worship the true God, or an idol of their own making? Yes, it does make a difference which view of God one accepts. Think of Jesus in the book of John confronting the Jews:
John 8:41-47 You are doing what your father did.” They said to him, “We were not born of sexual immorality. We have one Father—even God.”  Jesus said to them, “If God were your Father, you would love me, for I came from God and I am here. I came not of my own accord, but he sent me.  Why do you not understand what I say? It is because you cannot bear to hear my word.  You are of your father the devil, and your will is to do your father’s desires. He was a murderer from the beginning, and has nothing to do with the truth, because there is no truth in him. When he lies, he speaks out of his own character, for he
is a liar and the father of lies.  But because I tell the truth, you do not believe me.  Which one of you convicts me of sin? If I tell the truth, why do you not believe me?  Whoever is of God hears the words of God. The reason why you do not hear them is that you are not of God.
These Jews claimed to worship God – Jesus told them they were of the devil. They may have lived very pious daily lives, it did not matter.