...that one could read and sing this text as a rebuttal to the Church's Petrine authority...
Ephesians 2: 19 Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God, 20 having been built on the foundation of the apostles and prophets, Jesus Christ Himself being the chief cornerstone, 21 in whom the whole building, being fitted together, grows into a holy temple in the Lord, 22 in whom you also are being built together for a dwelling place of God in the Spirit.
Where I grew up, it was sung, intended, and expounded in sermons about how it is specifically a rebuttal against petrine, papal, and traditional authority.
Heretics and schismatics are not part of the Church, although the hymn implies that.
What do you think of "O Christ the Great Foundation," which is also sung to AURELIA. The author was Timothy T'ingfang Lew (I think he was Presbyterian). It is in Worship, 3rd edition. My parish sang this hymn when we had that hymnal. I liked the 4th verse, but we rarely sang 4 verses of anything at the time.
https://hymnary.org/text/o_christ_the_great_foundation/
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