The only circumstance where I have seen this situation is when the curve has been locked. While I cannot see the lock relationship on your curve in your video it still may be there because you have relationship handles toggled off. If you had set the area to record when recording your video to Solid Edge Application I would be able to confirm this hypothesis.
See the attached video in which I display the relationship handles, select the lock relationship icon and delete it allowing me to edit the curve.
Something else to watch out for is having "Perspective" on your model/sketch view...this can cause this problem to.
Design Manager Streetscape Limited
Solid Edge ST10 [MP2] Classic [x2 seats]
Thank you for the video and suggestions. Attached is a video with larger view. As you can see I'm trying it with Maintain Relationships and Relationship Handles on and off. I'm not sure how your getting that little box to come up.
I suspected it was happening because the sketch wasn't in edit mode but It's not the case.
Anyways, try this : select the sketch in the tree, right-click + lock lock sketch plane and try to move the point.
Thanks Laurenj, but didn't get it moving.
Messed around with it's now moving, I added and removed a point, held down the shift button and seems to be fine now.
Thanks again for the help.