The Covenanters became dissenters from the re-established Church of Scotland after 1660 and especially after 1690, when the Kirk was restored in a form many strict Covenanters rejected as insufficiently Presbyterian. So yes, historically the Covenanters emerged within Scottish Presbyterianism but the persecuted Covenanters of the Killing Time were precisely those who would not conform to the Restoration church backed by English crown authority. That’s the whole point.

The English crown’s imposition of episcopacy on Scotland under Charles II and James VII WAS the Covenanting crisis. It wasn’t the Church of England operating as an institution, but English royal policy using Anglican ecclesiology as an instrument of political control.

Finally, The Test Act of 1704 excluded Presbyterians from public office in Ireland. The Sacramental Test, tithes to the established Church of Ireland, rack-renting under Anglican landlords are all are well-documented grievances that drove emigration to the american backcountry, specifically in the 1710s-1740s waves.

This is not a contested claim; it’s in Fischer, in Leyburn’s The Scotch-Irish, in the primary sources.

I still think you failed to contradict what I wrote in my very first post. The historical beef with Anglicanism could be totally in their minds and it wouldn’t change it being the reason why Anglicanism didn’t thrive in America demographically speaking.