This is just a quick note to save others some searching in case they end up in the same position I've just been in. I span up a new Partner Developer Edition, hammered out some code, created some test code to cover it and then went to create an unmanaged package... and this is where I got stuck.
New orgs are created with Lightning Experience (LEX) enabled by default, and since I've been trying to force myself to learn to use it, I didn't immediately switch to classic mode. Now I'm fully aware that there is classic functionality missing from LEX, I just hadn't realised that this extended to the setup screens since they seem for the most part to be iframed, or at least incorporated in a similar manner.
It turns out that packaging is only available under the Salesforce classic UI. So if like me, you can't work out why it's not accessible and you're checking your edition features etc., consider switching to classic mode first. Progress is painful sometimes.