Public linking has not gone away, it is just different now, as far as I can tell.
This should work.BTW the link is an old flash ship preview I made for my own mod, it is no longer on the Public folder and it is now in an Starsector folder, inside Dropbox.
It just the public folder what's gone, and the ability to host media and display it directly on a webpage. Previously what I linked would have been opened in a new tab, now it won't, and it will just prompt you to download it, the same happens with images.
You can now share publicly any file, from any folder, it won't let you use dropbox as a image host though... Unless you're a pro or business user, but those users will also lose the ability to do that by September.
You can still use Dropbox to distribute mods (if the previous link works), not sure if it will work with version checker though, since it probably won't let you access the file directly.
So far I've been trying to use Google Photos as a image host. Seems to be working, at the very least I can open stuff with different browsers...
To do so I just made a new album, uploaded some images, opened one, clicked share, copied the link, and used this
tool to get a link to something that can be embedded, sadly it is a much more cumbersome process. If this works the image in the spoilers should be visible. Otherwise time to search for a different image host.