I realise with this question I was overcomplicating a really simple issue.
My question is really this:
How to have a page listing all items of a given custom post type?
Say the custom post type is foobar, I guess I want to be able to have a URL mydomain.com/foobar which will list them all. No categories involved. And with a template file in PHP which I can entirely customise.
Rather than edit the earlier question I have left it there in case the answers there are useful and applicable should someone need a "page without a page". But cybmeta is right, in the comments on that question, all I need in my case is an archive page.