JOEL - I will address a few of the scriptures that I believe you are refering to:

"Ye have not chosen me, but I have chosen you, and ordained you, that ye should go and bring forth fruit, and that your fruit should remain: that whatsoever ye shall ask of the Father in my name, he may give it you."(John 15:16)

Jesus did ordain the apostles to go out and bring forth fruit. The question is what does the fruit represent and what does the word "remain" mean?
Scriptures like these are subject to various interpretations. Does this scripture really say that their fruit "will" remain or only that it "should" remain. Jesus is not telling them what will deffinately happen, He is only telling them what He wants them to do. The things that God tells His servants that He wants done is always subject to the agency of man.
God told Moses that He would rescue His people from the Egyptians. He gave them His commandments and said He would lead them to a land flowing with milk and honey. These were supposed to be His chosen people. But because of their lack of faith and unrighteous behavior, he had to take back His commandments and delay their arival to the promised land for forty years.
In many cases human actions and choices can alter, postpone, or prevent the fulfillment of God's prophecies(See Jer. 26:12-13, 17-19).
The LDS church claims that, because of man's agency, there was an apostasy from the full truth of the Gospel and a loss of priesthood authority and church organization as Jesus taught it. But that doesn't mean that some of His teachings did not endure. A few of the fruits of the Apostles did remain on the earth.
Their scriptures and a basic knowledge of Jesus Christ and some of the things He represented prevailed throughout time. Unfortunately much of it was distorted and/or lost soon after the death of the Apostles, because of the weakness and agency of man.
This is the only way that I can understand this scripture and still keep it in harmony with other scriptures in the Bible about an apostasy and a restoration that was predicted by Jesus and His apostles. (Matt. 24:4-13, 2 Thes 2: 2-3, Acts 20:29-30, 1 Tim. 4:1-3, Acts 3:20-21)
I guess I could turn your question around and ask "If there was really NO apostasy, how can one explain these verses that prophesied that there would be one?"
The simple fact that there have been so many different Christian churches is evident that there was a loss of the pure, simple, and single Gospel of Christ. Paul said:

"One Lord, one faith, one baptism, " (Eph. 4:5)

Another scripture:

"For where two or three are gathered together in my name, there am I in the midst of them." (Matt. 18:20)

God's spirit can be with anyone regardles of their religion if they are sincere in their worship of Him. If that were not the case our missionary efforts would be a waste of time. What we have learned by revelation from God, however, is the importance of priesthood authority and performance of saving ordinances that are not found anywhere else on the earth today, except within His authorized church.
We have prophets that can properly interpret scripture so we can know without question what God is saying to us.
The word of God does last forever, but throughout time man has not always had complete access to it or a correct understanding of it because of their own actions and lack of belief.
See also:
http://www.mormonhaven.com/ben2.htm
http://www.mormonhaven.com/daryl.htm