It works fine when I do it.

Here's how I do it. (assuming recent version of Word with the Ribbon)

To insert another reference to an existing endnote:

In the Ribbon, choose References tab.
In the Captions section, select Cross-Reference.
In the Cross-Reference dialog box, select Reference type: Endnote, and select Insert reference to: Endnote number.
Choose the endnote you want from the list below that.
That will insert the endnote number. (Unfortunately, it's not superscripted; you'll have to do that manually. On the Ribbon, choose the Home tab and the Font section. Superscripting is there.)

And when I deleted endnote [1], all the references to endnote [2] turned into endnote [1], and so on, for all the endnote renumbering.