Ok,
So I create a custom shape.
I add points to the existing shape by alt+clicking an existing line.
As you describe I get a point with handles.
If I shift click it collapses the handles.
After they are collapsed it seems a shift double click removes them. It’s a little clunky though it seems like you do get some mis-clicks but that may be a UI limitation or something.
If the shift click then shift double click doesn’t work you could also pull the handles out position the point where you want then shift click to collapse the handles as a workaround.