Where does it say that you must believe in apostolic succession to be saved?
The real issue is the meaning of salvation. The Apostles taught something different than what is generally believed today. Salvation is not something that happens at one point in time, for all time. The Truth of the gospel was entrusted to the Apostles, and from them, to their sucessors. What is necessary for salvation is the Truth. Once the message is separated from the Apostolic succession, it quickly becomes warped. In some venues, it is practically unrecognizeable.
What Christians believe is necessary for salvation:
According to whose standard?
How come some Christians believe differently than you? If they see it differently, are they not really Christian?
2 Timothy 3
15and how from infancy you have known the holy Scriptures, which are able to make you wise for salvation through faith in Christ Jesus.
This is a reference to the Septuagint, rejected by Protestants.
Romans 1
16I am not ashamed of the gospel, because it is the power of God for the salvation of everyone who believes: first for the Jew, then for the Gentile.
This is a reference to the Apostolic kerygma, protected within the Apostolic succession.
Romans 10
9That if you confess with your mouth, “Jesus is Lord,” and believe in your heart that God raised him from the dead, you will be saved. 10For it is with your heart that you believe and are justified, and it is with your mouth that you confess and are saved. 11As the Scripture says, “Anyone who trusts in him will never be put to shame.” 12For there is no difference between Jew and Gentile—the same Lord is Lord of all and richly blesses all who call on him, 13for, “Everyone who calls on the name of the Lord will be saved.”
This is a reference to the profession of the creed, which was done at baptism. Baptism is the normative means by which people are added to the Body of Christ. A profession of faith is always made at baptism (calling upon His name).
Ephesians 1
13And you also were included in Christ when you heard the word of truth, the gospel of your salvation. Having believed, you were marked in him with a seal, the promised Holy Spirit, 14who is a deposit guaranteeing our inheritance until the redemption of those who are God’s possession—to the praise of his glory.
This is another reference to baptism, during which the individual is sealed by teh HS. The Gospel of salvation was committed to the Church, and people acquired it through the Church.
1 Peter 1
8Though you have not seen him, you love him; and even though you do not see him now, you believe in him and are filled with an inexpressible and glorious joy, 9for you are receiving the goal of your faith, the salvation of your souls.
Those who believed came to do so through the preaching of the Apostles, and those they appointed. This faith is the fruit of the preaching through the Apostolic Succession.
John 3
5Jesus answered, "I tell you the truth, no one can enter the kingdom of God unless he is born of water and the Spirit.
Another reference to baptism, administered by the Apostles and their successors!
John 14
6Jesus answered, "I am the way and the truth and the life. No one comes to the Father except through me.
The Church is the means by which Christ has chosen to manifest Himself in the world, and through which we come to Him, and thus, to heaven.
2 Corinthians 7
10Godly sorrow brings repentance that leads to salvation and leaves no regret, but worldly sorrow brings death.
It is interesting that you included this one. I have had Reformed Christians on this forum tell me that it is not necessary to repent of one’s regrettable deeds in order to be saved.
2 John 1
9Anyone who runs ahead and does not continue in the teaching of Christ does not have God; whoever continues in the teaching has both the Father and the Son.
The teaching in which we are to continue is that which was entrusted to the Apostles, and from them to their successors. The way this teaching was considered valid was if it came through the Apostolic Succession, rather than teaching from any other source. The Teaching is also called “the Way”, in the NT, and it is the lifestyle of a Catholic.