Martin Luther...actually quoted, before his death, saying "There are almost as many sects and beliefs as there are heads; this one will not admit baptism; that one rejects the Sacrament of the altar; another places another world between the present one and the day of judgment; some teach that Jesus Christ is not God. There is not an individual, however clownish he may be, who does not claim to be inspired by the Holy Ghost, and who does not put forth as prophecies his ravings and dreams."
With self-interpretation of the Bible, and you come to a different interpretation than the churches in your area, nothing can stop you from making your own church. Nobody has the authority to say you are wrong in your interpretation because that would then place them at the same level of authority has the Bible. Which is against SS.
With SS, everybody is right in their interpretation of the Bible, and everybody is also wrong in their interpretation of the Bible.
Logically, since not everybody is right in their interpretation of the Bible, there needs to be an authority higher or equal to the Bible to claim what is the correct interpretation.
I'd like to know, where did this presupposition come from that different sects/groups/denominations happen because of differing interpretations of God's book?
(I noticed that Martin Luther, in your quote, doesn't mention interpretation at all.)
In my experience, fringe groups start when people get certain ideas and then try to shoehorn them into Scripture. The Church of Christ's ban on all musical instruments is a perfect example of this. What verse did they misinterpret to get this idea? No, rather they got the idea first--from where, I'd like to know--and tried to wrench Colossians 3:16 out of shape in order to support it. A verse which makes not a mention of said instruments.
Speaking of Colossians, Paul says this in chapter 2 of that book:
Let no one cheat you of your reward, taking delight in false humility and worship of angels, intruding into those things which he has not seen, vainly puffed up by his fleshly mind...
This reminds me of things like the Jehovah's Witnesses claiming that only a select 144,000 are going (all right, have gone) to heaven. They cite Revelation 7 as proof. It doesn't mention it--and how, reading it, would a person ever get that idea?
Catholics arguing against SS seem to be pushing the idea that the Bible is like a Rorschach test, that if ten people read it for themselves, they'll come away with ten different interpretations. That hasn't been my experience reading it over the years. Peter mentions some of Paul's letters that are hard to understand, and the book of Revelation is a fireworks display of symbolism; but those are the exceptions, not the rule. The main things are the plain things, and the plain things are the main things.
Our Creator, who wishes for all to be saved, wouldn't make his revealed word so difficult to understand that only a select few can interpret it correctly. Why would he do that?