That fixed it. Hitting download on the top of the page (the download button that shows up after claiming) now lets me download the full version.
I also double checked in the itch client (which I hadn't tested before) and it lets me download the full version there too.
Now I'll know exactly what to tell devs in the future if they mistakenly think they're giving away a game permanently. And they won't have to try multiple things to see what works like you did.
Edit: had you asked your friend if they could still download the full version before or after you added the demo back to the page? If it was before you had added the demo back then that would be why they could still get the full version.
But if they checked after the demo had been re-added, then I have no idea why they would have been able to still download the full version. Afaik if would have been re-adding the demo that made it only let the demo be downloaded again, not the sale ending by itself.
But like I said it's fixed now with that "paid with a free file" setting that you changed.