In what sense do Mormons believe salvation is only through Christ?

This answer is in response to edits and clarifications on the original question, and therefore my original response is to the question as it originally stood.

The Book of Mormon teaches:

All things denote there is a God; yea, even the earth, and all things that are upon the face of it, yea, and its motion, yea, and also all the planets which move in their regular form do witness that there is a Supreme Creator

-Alma 30:44

This passage explicitly indicates that everything which we can see was created by a Supreme Creator. Mormon theology teaches that God is the father of all the spirits of mankind. This is echoed in statements from the church:

One of life’s great questions is “Who am I?” A beloved Primary song helps even little children answer this question. We sing, “I am a child of God, and he has sent me here.” The knowledge that we are children of God provides strength, comfort, and hope. You are a literal child of God, spiritually begotten in the premortal life. As His child, you can be assured that you have divine, eternal potential and that He will help you in your sincere efforts to reach that potential.

-True to the Faith, "God the Father"

It is clear that in Mormon theology, all things observable to us and all humankind are the offspring of a single entity, God. Additionally, the LDS church teaches the God answers the prayers of all of his children in every religion. Joseph Smith also taught:

Could man exercise faith in God so as to obtain eternal life unless he believed that God was no respecter of persons? He could not; because without this idea he could not certainly know that it was his privilege so to do, and in consequence of this doubt his faith could not be sufficiently strong to save him.

-Lectures on Faith, 3

It seems clear from these contexts that Mormon belief is that there is one God who is the God of all which is in the world, and that other religions worship the same God from a Mormon perspective, albeit with some perceived erroneous notions.

For this reason, Mormons spend energy on proselyting. Not to convert them from one god to another, but to teach them the truth as Mormons believe it. To extend the analogy: All that we see is one single department, everyone we know works for the same Boss; there is none other. He is not a respector of persons. We are only aware of the potential existence of other Departments, but know absolutely nothing about them. Our Boss and our department is the only one we worry about.

Mormonism receives much criticism for the doctrine of the plurality of gods, perhaps with good reason. However, there is widepread ignorance as to how it is interpreted inside Mormonism itself. There is several orders of magnitude more Mormon-produced literature on the nature of God, Christ, and our relationship with them than there ever has been on a plurality of gods.

What I'm going to write here is how I understand this. Mason Wheeler, please tell me if anything here seems to be inaccurate.

We are spirit children of God. Jesus Christ is also one of God's spirit children, but he became perfect like His father before mortality. Christ was able to pay the price for the sins of those who will repent and was able to be resurrected so that everyone who is born can be resurrected.

Because of Christ, we can eventually become perfect and have immortal bodies like God does if we repent. We have the potential to become like our father. Also, God once was in a state like we are now in.

It would seem to me from this that there are probably many other gods in this family tree. However, we do not need to know any details about them to accomplish our missions in this life. These gods are not the gods of other religions on earth, nor are they the gods of The Church of Jesus Christ of Latter-Day Saints.