The Scriptures are a sufficient standard to remove error from the church, but they are not the mechanism by which error is removed from the church. That's true whether one's ecclesiology is papal, patriarchal, (some other form of) episcopal, presbyterian, or congregational. I realize that those today with some of those ecclesiologies don't use the Bible that way, and that is their loss.
The Bible should be the standard of judgment within the church, yet we must distinguish. The Bible is the rule of faith, but it is not the person who applies that standard or rule to the situation. In other words, one of the functions of the church is to serve as a judge of controversies and to apply the teachings of Scripture to the matter at hand. In this way, churches can oppose heresy.
The churches who properly use Scripture do not simply wait for a Bible to zap heretics with lightening bolts, they search the Scriptures to see whether the person is teaching something that is contrary to the Word of God.
This function of Scripture as a sufficient rule of faith is not a new function. It is not something that the Reformers dreamed up. It is not even something that the apostles dreamed up. Even the mighty Ezra was not the originator of this principle of the sufficiency of Scripture as a rule, to which nothing needed to be added.
Deuteronomy 4:2 Ye shall not add unto the word which I command you, neither shall ye diminish ought from it, that ye may keep the commandments of the LORD your God which I command you.
Every word of God is pure: he is a shield unto them that put their trust in him. Add thou not unto his words, lest he reprove thee, and thou be found a liar.
God gave the people of Israel Scriptures. He expected them to use those Scriptures as a rule, and he criticized them for adding human traditions to them:
But he answered and said unto them, Why do ye also transgress the commandment of God by your tradition? ... And honour not his father or his mother, he shall be free. Thus have ye made the commandment of God of none effect by your tradition.
For laying aside the commandment of God, ye hold the tradition of men, as the washing of pots and cups: and many other such like things ye do. And he said unto them, Full well ye reject the commandment of God, that ye may keep your own tradition. ... Making the word of God of none effect through your tradition, which ye have delivered: and many such like things do ye.
The Sacred Scriptures, even of the Old Testament were, you see, a sufficient standard. That's why Jesus appealed to them:
John 5:39 Search the scriptures; for in them ye think ye have eternal life: and they are they which testify of me.
And why the Bereans were commended for judging Paul's teaching by them:
Acts 17:11 These were more noble than those in Thessalonica, in that they received the word with all readiness of mind, and searched the scriptures daily, whether those things were so.
That's why when the apostles wanted to decide a matter they turned to them and applied them to the matter at hand:
And after they had held their peace, James answered, saying, Men and brethren, hearken unto me: Simeon hath declared how God at the first did visit the Gentiles, to take out of them a people for his name. And to this agree the words of the prophets; as it is written, After this I will return, and will build again the tabernacle of David, which is fallen down; and I will build again the ruins thereof, and I will set it up: that the residue of men might seek after the Lord, and all the Gentiles, upon whom my name is called, saith the Lord, who doeth all these things. Known unto God are all his works from the beginning of the world. Wherefore my sentence is, that we trouble not them, which from among the Gentiles are turned to God: but that we write unto them, that they abstain from pollutions of idols, and from fornication, and from things strangled, and from blood.
Just because God gave the apostles and after them the elders does not mean that there is any guarantee that the churches will always stick to the rule of faith. After all, there were elders in the Old Testament era, and God ordained not only those rulers but Kings, Prophets, and Priests as well.
And yet Israel, the only visible congregation of God fell over and over again. The Scriptures were sufficient, but humans err. There were folks like Naboth who were wrongly condemned by a sinful application of God's law, and there have been men like Galileo and Hus who were wrongly censured by people who profess to be Christ's followers.
We are not free from the risk of human tradition in our churches. Paul wrote to warn us of this:
Colossians 2:8 Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ.
And Peter makes the parallel between the Old and New Testaments even more explicit:
2 Peter 2:1 But there were false prophets also among the people, even as there shall be false teachers among you, who privily shall bring in damnable heresies, even denying the Lord that bought them, and bring upon themselves swift destruction.
So then, let us heed that warning and recognize that the sufficiency of Scripture to decide all matters of faith and morals as the standard of judgment is one thing. The application of that standard to life is another. Humans will err, but God's word remains powerful and infallible:
Isaiah 55:11 So shall my word be that goeth forth out of my mouth: it shall not return unto me void, but it shall accomplish that which I please, and it shall prosper in the thing whereto I sent it.
N.B. Many thanks to Steve Hays for drawing the sufficient standard / sufficient mechanism concept to my attention.