In addition to my Desktops, I am using Everdo on my mobile phone. First of, I think it is a great app.
I am running a pure AOSP build without play services. The app itself works perfect also without any google services, which is awesome.
So currently I am downloading the builds from the forum, where i need to keep track an check for updates manually.
I think it could fit a great part of the actual target audience when providing a dedicated F-Droid repository. This way, we could have automatic updates, instead of the manual untrusted install.
As far as I know, only open source apps are allowed there. From the inclusion policy:
All applications in the repository must be Free, Libre and Open Source software – for example, released under a GPL or Apache license. Every effort is made to verify that this is actually the case, both by visual inspection of the source, and by building the application from the published source. If in doubt about a license, please refer to the GNU license list.
For software to be FLOSS, the software in its entirety must be so - including all libraries and dependencies used. Additionally, it must be buildable with only FLOSS tools.
The inclusion policy applies, if an app shall be added to the “F-Droid default repository”, which is maintained by F-Droid. In this repository, only open source software is added, when it fit mentioned criterias.
Custom repos are another thing: They are not maintained by F-Droid and could contain also binaries.
From the Repo setup guide:
Custom repos do not even need to build the APKs, they can just be “simple binary repos” of any APKs.
The only drawback is, that you would need to host this Repo (basically anyone could host one) and add the Apks to it.
From a Users perspective, adding a F-Droid repo is very easy. Basically just sharing a link, or scanning a qr-code. It is described here.