“Joseph Smith said that all other churches had been corrupted (I think all churches have said this at some point) and that without the restored gospel of Jesus Christ, there was no salvation.”
Alright, now if the Jesus who Mormons believe in is the same Jesus that all the other Christians believe in, then why do we require the Mormon gospel for salvation? What requirements are laid out in the Mormon gospel which you believe, I suppose, have been removed or altered from the gospel that was given to the rest of us? In other words, is Christ sufficient to save us, or do the Mormons teach that we need “Christ plus”? (Christ plus baptism, Christ plus works, what have you)
Actually a good question, I'll answer it three fold.
1) The trinity. If we are correct about the nature of God then Orthodox Christianity is teaching a heresy. If they are right, we are teaching heresy.
2) A) Authority if the Catholics are right and they have unbroken authority coming down to today from Peter, then We and all Protestants are heretics and need to repent and be brought into communion with the Catholic Church.
2) B) Authority In every instance throughout the Bible when the Gospel was corrupted, God sent men called by him to re-establish his church, we call them prophets. If the Catholic Church lost the authority somewhere along the way (Bogias for example) then only a church established by a prophet of God has the possibility of having the authority to act in God's name as the Bible requires for Baptism. The Mormon church is the only church that meets this criteria. Protestants don't have an authority leg to stand on and are not a consideration, authoritatively.
3) If you are not baptized into the church of God by the authority of God, then your soul is in mortal jeopardy
Conclusion: If there is even a possibility that one is a member of the wrong church, they should check it out. I attended a Catholic Bible school (until asked not to attend anymore - I was not misbehaving, just a Mormon, I'll tell the story if you want) I have spent much time in prayer and supplication to know God's will for me. I prayed about the Book of Mormon, got an answer, it was accompanied by a testimony of Jesus Christ as my Lord and Savior. I encourage all men to get a book of Mormon, read it, pray about it, and see for themselves if God says it's true or not. Don't believe me, don't believe anyone else, don't believe any man no matter how learned in a matter so important. If you can read a book, pray about it and know for yourself, why wouldn't you? You don't have to be smart, strong, or popular. God answers sincere prayers. God will answer his children's sincere requests for knowledge.
Thank you for asking this.
God bless and keep you
Delph.