Not exactly answering your question but you can use the app Obtainium to fetch the apk URL from a website/github repo and many other sources to install directly. It also supports fdroid repos and many other sources out of the box. Kinda half way what you mentioned in your first paragraph.
Of course Github is just an example but you can pretty much regex any URL and further filter out anything in order to get the apk link with it. So depending on your level of privacy requirement and trusted sources, you can skip all the centralized ones and build your own list of sources.