It appears anchoring objects to a page locks the objects to that exact page number.
Yes, that is exactly what "Anchor to Page" does.
If you want something to be positioned relative to the page, but not to the specific page number, you have to use one of the other anchoring options.
In the positioning configuration for the frame, you can specify the position relative to the page, if you want.
Note that using the mouse to position things is incompatible with an exact position relative to the page (or anything else, other than the anchor itself). If you want to position something relative to the page, you always have to adjust its position through the properties dialog. If you use the mouse, Writer will change the position back to anchor-relative coordinates.
Drives me nuts.