The question seems to be whether they are bound to Scripture as they understand it now, or Scripture and the reasonable moral inferences therefrom, as these have been understood in continuity with the whole Christian tradition for 2,000 years.
No, I know something of the Presbyterian history in the US. The modernists staged a coup in the early 20th century. Many of the much more conservative splinter groups are a product of that cataclysm. Gary North has written an extensive history of this called “crossed fingers,” in which he chronicles how those who had adopted the modernist premises cheated by presenting themselves to the seminaries as faithful to the Westminster Confession, but who in fact engaged in the deceit of “mental reservation,” crossing their fingers behind their back, as it were, and not revealing their full rejection of Biblical authority until they were well established in the woodwork of American Presbyterianism.
This practice, BTW, is something every good Catholic should recognize as familiar, through their own beloved Cardinal Newman, whose Oxford Movement used it extensively to move surreptitiously among English Protestants until it became safe to reveal their true meanings toward Rome.
Bottom line, the current Presbyterian mainliners in America long ago ceased being Sola Scriptura fellowships, and rather have been following a long path toward ecumenical paganism. I’ve been in these churches. I know whereof I speak. If anything, their problem is their authority doctrine is rooted in the same man-centeredness as Rome. Present them with Biblical authority, and they don’t go off to some variant translation or interpretation. They just look at you cross-eyed for daring to question modern liberal orthodoxy with that old, irrelevant book. Seriously. One fellow my dad met in the modernist Methodist church ground a Bible under his knee in a pew to make the point clear to my dad, many years ago. That’s as anti-Sola Scriptura as it gets.