I’m always tempted to share the non-published version of docs to avoid this, but I don’t really want people poking around in the hidden pages that I use to update and manage the doc.
Coda, I think the best way for you to publicize Coda is to let us make the best possible docs, and the floating paraphernalia seem pretty retro, in my opinion—like those old social share buttons that used to appear on a bunch of blogs.
This is gone if the doc is published on a custom domain or subdomain and you access it like that and not through the coda.io/@vanity/link. So unless you already have one, you can get a free or dirt cheap domain and connect it to your doc.
The other option is to serve the doc through Cloakist but that’s the 3rd party service that’s gonna cost the extra $10/mo.
Other than these, I don’t know of a way to remove that block from a published doc.