OK, I think I’ve just figured this out.
I went to the diagnostics section of the plugin and noticed that PHP curl support wasn’t enabled. I had noticed this previously, but as the homepage was being downloaded OK, I assumed it wasn’t important.
When I installed php-curl on my server, it started working properly!
The plugin has now exported all the pages of my site 🙂
I guess the key is to look into the diagnostics page and ensure everything is marked as “OK”.