The section has the purpose to praise the supporters of King David, who, in the system of parallelisms between Israel and England that structures Absalom and Achitophel represent King Charles II. The fact that Dryden praises these figures makes explicit his support for Charles II and for the conviction that the instituted authority of kings should not be overthrown. People and political leaders cannot simply decide to change the order of the succession as Achitophel (Earl of Shafetsbury) would like to do by putting David's illegitimate son Absalom (Duke of Monmouth) on the throne.
The lines you refer to in your questions cite important institutional figures under the usual Biblical disguise that supported the power of the King. They realized that if the order of succession was to change, the nation could dangerously come to a state of chaos. Barzillai represents the Duke of Ormond, while religious insititutions are represented by Zadoc (the Archbishop of Canterbury) and the Sagan of Jerusalem (the Bishop of London). Political institutions are represented by Adriel (the Earl of Mulgrave), Jotham (the Marquis of Halifax), Hushai (Laurence Hyde), and Amiel (Edward Seymour). The section thus shows both political and religious institutions as countering the rebels and supporting the actual King. Such support makes David act against the rebels, changing his initial attitude of leniency because of his benevolence for his beloved Absalom.