Your fourth paragraph I think is the key to your problem and you've already solved it yourself.
If your points don't have z values to start with and aren't draped onto your surface, then the values (2, 130) you're using are only above their nominal height of 0. If your terrain is higher than that, then yes, all your sight lines are underground. The reason it's working with the button is that your click is registering at the surface and not the point's elevation. Your offset does need to be in addition to the point's elevation - be that because the points are draped on a surface or because they have an elevation to start with.
The simplest fix is probably to get your points to 3D, which there are a couple of options for at that help file link. You'll probably want to go that route, as I'm not sure using the drape method would yield you anything if you went to a model - it would be better to have the discrete values in the data.
Note that having a z value field and having z-enabled data are two different things. When you create a feature class, there is a check box to enable z values. This gives a z component to the shape field, and not just another field with a z value. Depending on the tools you're working with, this distinction is important. Note in the Construct Sight Lines help where it gives the priority that various height fields will be used (I think you've seen this based on your comment). I believe that Add Surface Information does convert to 3D (therefore giving a shape.z value) but I'm not positive on that. It's been a while and I'm not actively testing this, just going from memory.
ArcGlobe I'm not as familiar with. I do know there are differences between Scene and Globe in the way things are handled.