Sorry to reopen this again but after a bit more testing I see that Intersection snapping does work but just not the same as it did in ArcMap, it is not as inclusive or consistent and I would like to make a plea for the Pro snapping intersection tool to be revisited.
I would like to illustrate. As stated above, I wanted to snap to the intersection of two farm polygons that occur on the edge of the farm layer. In ArcMap, intersection snapping does snap to where these two polygons meet on the edge.
However in Pro the snapping tool does not identify this as an intersection point. In Pro intersection snapping seems to work best where two different snappable layers intersect, not polygons within the same layer. It can work but then it is strangely inconsistent and inaccurate from what I can tell. That is at least my experience.
From the tool help, it does say "Snap to the intersection of two or more line or polygons features when using interactive map tools" ... so it really should snap to polygons within the same layer.