This may sound a bit simple, but in user preferences—>Themes—>3D view Are the vertex and vertex selected colors the same?
EDIT
Ooops, missed the part about other objects not having the problem. That is strange. If you append the object to a new scene, is the problem still there?
Yes the problem still exists after moving the object to another file, I found that I can click them when I have the object transparent, but when it’s not maybe the object itself gets in the way of the vertex points.