You are right, you should be able to download your books. In fact, not only should you be able to download them, you should be able to download them in virtually any e-reader you have on your iPad.
I also have my books in the Public folder in Dropbox and use Calibre2OPDS (version 2.3 rev. 132) to create a catalog in the folder. When I open up my Calibre2OPDS catalog in Safari (have a link saved to the iPad's homescreen), I get the main catalog, which lists authors, tags, series and all books. When I press on one of the categories, say, author, I get another web page for each author (alphabetical). I select "A" authors, get another web page ... etc until I actually get to a book's thumbnail. At the bottom of that web page, there is a Downloads place. I can select to download the book and yet another Safari page opens, with two download buttons: "Open In..." (where you can select the reader you want) and "Open in iBooks (... which is actually whichever app you last used with the catalog).
So, it works great for me. Which doesn't help you any, because I don't know how to help you. So, I've attached 3 pictures to show you all the blocks and checks I've made in Calibre2OPDS when I'm getting it to create a catalog.
Maybe you can make yours match mine and that'll help? That's all I've got. Hope it works and good luck.
Marilyn
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