The problem with point mode against a point layer is the accuracy. When you are using the SelectionOnly mode, the selection is done against the server, and this is simply sending the coordinate you clicked to the server, and looking for any points that intersects that point. The thing is with point/point intersection that you have to hit EXACTLY at where that point is down to the last digit precision (note the intersection is done on the geometry, not the symbol which is a lot larger than an infinitely small point). In other words: Selection does work with point, IF you can click at the right point. However this is a good as impossible to do.
Bottomline: Do not use point as selection mode against a point layer.