Delete key does not delete notes in Preview

In Preview, when you want to delete an annotation, press ⌘+i to open the Inspector panel. The right-most tab in that panel is your ordered annotations. Click the one you want to remove, press the backspace key, and then click once in the PDF and the annotation will disappear.

When a note becomes active, it is in edit mode, so the delete key is acting on the text.

You have to move the cursor to the border and click on the border with the hand cursor. You can then delete the note.

If the insertion carat blinks, then you are back in the text editing mode.

I am on macOS 11.2.1. I open an existing PDF in Preview, add a Note annotation, watch it appear on the document icon, and then manually save the PDF and exit Preview. Upon opening the same PDF in Preview, I can remove the Note annotation as I described earlier. I cannot however duplicate your circumstances as you have described.

It works when I do it. I can't explain why yours is different. It is certainly a difficult process to get right.

If this is a PDF that you already saved as PDF via the print menu, or exported to PDF previously from Preview, then that note annotation will not be removable from the PDF.