Yes, node snapping only works with the Node tool (and "Snap to object geometry" enabled) (similarly as e.g. in Illustrator and CorelDRAW). With the Move tool, only snapping to shape key points works, and I think that is also limited to bounding box points.
Boolean operations are the way to go whenever possible. I did this unnecessarily complex way because the shapes' bottom join points where not merged, so I deleted the vertical segments and joined the bottom connecting nodes and closed the curve at the top, which often requires nudge operations to get the nodes overlapping without distorting the shape (joining by dragging with snap aid typically does that), and closing (and removing possible double overlapping nodes) with the Divide boolean operation.
@haakoo made it look like a trivial problem, which it was, in the end, as the bottom nodes already were fully aligned, and boolean operation was only one alignment operation away from being available.
The Curve snapping: nodes and control handles video tutorial explains how each of the Node Tool's "Snap" options on the context toolbar work. As mentioned at around 0:40 in the video these snapping options are completely independent of the "global" snapping options on the main toolbar.