... I wasn't seeing the familiar Page Settings page. I was seeing this:
No sign of my picklist of Page Templates at all. I knew I had encountered this problem before, but couldn't remember what the solution was, so I got on to trusty Google, figuring someone is bound to have the answer to this vexing issue. But no ... nothing doing.
I tried going to the Pages Directory and seeing if Editing Properties of the page would help, but that got me nowhere.
Then, running out of options, I fired up SharePoint Designer and navigated to the sub-site. And there it was!
The dad-blasted Page Layout was detached, obviously from some earlier Designer fiddling and I'd forgotten to reattach. Simply reattaching the Page Layout fixed the problem and I was able to access the Page Layout picklist in the Page Settings area.
So hopefully, this will save someone thrashing around, trying to figure out why they can't change the Template for a SharePoint Publishing page.