Absolutely. A Bible is
not even required reading.
Romans 1:20
For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse.
Psalm 19
The heavens declare the glory of God, and the sky above proclaims his handiwork. Day to day pours out speech, and night to night reveals knowledge. There is no speech, nor are there words, whose voice is not heard. Their voice goes out through all the earth, and their words to the end of the world.
Psalm 8:3
When I look at your heavens, the work of your fingers, the moon and the stars, which you have set in place,
Psalm 8:1
To the choirmaster: according to The Gittith. A Psalm of David. O Lord, our Lord, how majestic is your name in all the earth! You have set your glory above the heavens.
Absolutely.
I believe there is a considerable difference between being born again, and being saved. The former must come before the latter is ever possible.
Being born again, as the above scriptures make abundantly clear, does not require a theology of the cross. It just needs a man to turn his heart to God, the God who reveals himself to all men!
Should he die, anyone who is born again will be received by God, because he has already passed from death to life. But after New Birth, Salvation is an ongoing process that lasts a whole lifetime, although sadly, most remain spiritual infants till the day they die.
I was born again as a child, more than a decade before I heard the gospel. I heard God speak to me directly, among other things telling me that he lived in my heart, not in a church building. He also told me that eternal life was nothing to do with being a good person, but because of what he had done.
I knew nothing of the cross till ten years later.
So regarding the requirement to know a correct theology of the cross, here's what Paul says.-
1Cor1v18 For the message of the cross is foolishness to those who are perishing, but to us who are being saved it is the power of God.
Unless and until a person is born again, he is classed as spiritually dead, ie. "perishing!"
But look above, the message of the cross is "foolishness" to such people, so how can an unbeliever ever possibly understand what Jesus did on the cross? Clearly, comprehending the message of the cross is only possible to those who are already born again. Therefore being "born again" is possible before knowing what Jesus did on the cross.
Therefore, anyone who turns to God can be born again, whether Trinitarian or not!
So what of the cross?
All new life, all salvation, all forgiveness of sin, depends on the price Jesus paid. This is true for Adam himself, for Noah, Abraham, Moses, for David, Hezekiah Joseph and even Mary.
Jesus is the God we see all through the OT, including He who walked with Adam Abraham and Moses. Therefore when Jesus speaks of believing in Him, it is the same as believing in Jehovah, El Shadai, Elohim, etc.
I shall end here, but there is of course much more to this line of thinking.
The Jesus of the LDS is the spiritual brother of Satan, and appeared to Joseph Smith and commanded Mormonism to be established because all other Christians churches and denominations were abhorrent to the real God, who is himself a created being.
