Reading the Scripture "
publicly to the congregation" is a part of conducting the public worship of God, and therefore it is to be done only by those who have been properly called to that office in the church. In the absence of an ordained minister or licentiate, the elders of the church may properly appoint sonic person to read the Scripture and conduct a prayer meeting or "fellowship meeting."
What the catechism denies is that any private Christian may lawfully take it upon himself to conduct public worship, without being appointed to do so by those whose office it is to rule the house of God.
Deuteronomy 31:9-13 (NKJV)
9 So Moses wrote this law and delivered it to the priests, the sons of Levi, who bore the ark of the covenant of the LORD, and to all the elders of Israel.
10 And Moses commanded them, saying: "At the end of every seven years, at the appointed time in the year of release, at the Feast of Tabernacles,
11 when all Israel comes to appear before the LORD your God in the place which He chooses, you shall read this law before all Israel in their hearing.
12 Gather the people together, men and women and little ones, and the stranger who is within your gates, that they may hear and that they may learn to fear the LORD your God and carefully observe all the words of this law,
13 and that their children, who have not known it, may hear and learn to fear the LORD your God as long as you live in the land which you cross the Jordan to possess."
Nehemiah 8:2-3 (NKJV)
2 So Ezra the priest brought the Law before the assembly of men and women and all who could hear with understanding on the first day of the seventh month.
3 Then he read from it in the open square that was in front of the Water Gate from morning until midday, before the men and women and those who could understand; and the ears of all the people were attentive to the Book of the Law.
Nehemiah 8:3-5 (NKJV)
3 Then he read from it in the open square that was in front of the Water Gate from morning until midday, before the men and women and those who could understand; and the ears of all the people were attentive to the Book of the Law.
4 So Ezra the scribe stood on a platform of wood which they had made for the purpose; and beside him, at his right hand, stood Mattithiah, Shema, Anaiah, Urijah, Hilkiah, and Maaseiah; and at his left hand Pedaiah, Mishael, Malchijah, Hashum, Hashbadana, Zechariah, and Meshullam.
5 And Ezra opened the book in the sight of all the people, for he was standing above all the people; and when he opened it, all the people stood up.
See also Volume 4 of Ridgeley's commentary on the WLC:
http://digitalpuritan.net/thomas-ridgley/