OK, I think I have a simpler test case and a better idea of what the actual problem is.
If you project the sketch origin into a sketch, I don't think you're supposed to be able to delete it. You're stuck with it forever. Not every impossible-to-delete point is a sketch origin, but all sketch origins should be impossible-to-delete points.
However, it seems you can de-magic the sketch origin so that it's neither undeletable nor constrained to be at the actual origin. Here's one way to do it.
1) In a new sketch, draw a line. Anywhere, any orientation.
2) Apply a Coincident constraint between the midpoint of the line and the sketch origin.
3) Draw another line that starts at the midpoint/origin. You can end it anywhere.
4) Use the Trim tool to remove the two halves of the original line.
At this point, you can drag the second line away to show an underlying point at the origin. I'm not sure if it's actually a de-magicked sketch origin or just a remnant of the first line's midpoint. However, you can move and delete that point freely and there is no other sketch origin. And you can no longer reference the real sketch origin in any way, short of explicitly turning on a coordinate system and projecting that in (which seems like a completely separate mechanism).
[Parenthetically, and probably unrelatedly, I notice that you cannot manually Project the sketch origin, which seems somewhat peculiar now that I think about it. You can only activate the sketch origin by referencing it implicitly. It doesn't seem to matter whether "Automatically project edges on reference" is turned on or off - the sketch origin always seems to behave as if that option were enabled.]