There is nothing dumb about using Word 2003. It may be boring, because it works.

Unfortunately, page numbers are truly not that simple.

Selecting and deleting does work. I do not have a command to Remove Page Numbers in Word 2003. AFAIK, that command originated in Word 2007.

For more, see: Page Numbers



Here is the help text on removing page numbers from Word's help:
Remove page numbers

1. On the View menu, click Header and Footer.
2. If you positioned the page numbers at the bottom of the page, click Switch Between Header and Footer on the Header and Footer toolbar.
3. Select a page number. If you inserted page numbers by using the Page Numbers command on the Insert menu, make sure to select the frame around the page number.
   How?
   1. Click the page number to make its frame appear.
      
   2. Move the pointer over the frame's border until the pointer becomes a four-headed arrow, and then click to see the frame's sizing handles.
      
      If the handles are not visible, you might have clicked the contents of the frame, not the frame itself.
4. Press DELETE.

Notes

• Microsoft Word automatically removes the page numbers throughout the document. To remove page numbers for part of a document, divide the document into sections and break the connection between them.
• If you created different first-page or odd and even headers or footers, make sure to remove the page numbers from each different header or footer.

As far as I can make out, you're saying I have to individually remove each page number. I've already done that for a couple of pages, but surely there must be a way that tackles the whole document in one swell foop

That's only applicable for page #s that have been inserted in frames (which Word 2003 typically used); not all are and, strictly speaking, deleting the frame isn't necessary (it's untidy to leave it behind, though).

__________________
Cheers,
Paul Edstein
[Fmr MS MVP - Word]

I agree. I was quoting the help section on removing page numbers.