Thanks Jeff for letting me take this.
Investigation shows a few key things that are all working together to make this experience.
1. A bug can be seen at precisely 1:07 in your video. When you drag the center of the ellipse to move it, the dot turns black, indicating it thinks it's fully constrained, which it's not, which is a bug and is now logged. When you then drag it to the origin, all the construction lines turn black indicating they are fully constrained. Nothing will drag when it's fully constrained.

2. There is an existing bug that is also bothering you. If you have a completely un-dimensioned ellipse and the center is constrained, you cannot drag the axis points but you can drag the curve. If you add a dimension to one axis you'll find you can drag the axis points. In your video, after adding the dimension, you did not try to drag the axis points, instead only dragging the curve. So while this is being fixed (which will take a few weeks at the soonest), simply add a dimension to your ellipse axis and you can drag the end points around. This is an incomplete workaround if you require the drag for both axis. If you could show us how that completely blocks a real workflow, i.e. in your model not just a sketch, I can ask for higher priority on the fix (no guarantees here, just making a case for you).
3. I logged an improvement that asks for sketch axis to be driven by angle dimensions. Currently you must add construction lines to your ellipse to give it an angle, see video below.
*For your workflow where the ellipse must be anchored at the center, you must apply the angle dimension and then use a coincident constraint. This is part of the bug case I mention above.
Phil Eichmiller
Software Engineer
Quality Assurance
Autodesk, Inc.