The poor handling of snapping is why I rarely use JOSM. The iD editor makes it very obvious when something is snapped. My guess is that the snap radius in JOSM is a fixed value that does not vary with zoom level. The only way I was ever able to snap to a node in JOSM was to zoom in really close. Even then I wasn’t sure until I tried moving the node. My productivity ground to a halt with it so I only use it for certain tasks.
I should note that I am a noob though so hopefully someone else could enlighten us.