I've no clue why it's not recognizing your login state. A JSONP request just sends an ajax HTTPS request to our site, but it allows cookies to work. Your login cookie is recognized and your login state is then factored into the conditional checks in our CB Package Builder usage. For whatever reason your cookie isn't being recognized.
Please keep in mind your site will cache Install from Web for 24 hours. If you were not logged in, it cached, then you logged in you would still see it as if you weren't logged in and will need to manually refresh it using the refresh link on the bottom right of the tab.
There's not much more I can suggest. We do have a Download History so if you download all the plugins your site is using you can redownload latest versions in 1 click from your download history below to at least make manual updating easier.
www.joomlapolis.com/downloads/download-history
Once we've a token system implemented there will be an alternative for those having issues with JSONP requests. It could be possible it's simply a security setting on your site, computer, or browser blocking it.