I'm not sure how it couldn't be a business. The App Store currently takes a 30% cut of every purchase. The business is running and developing an app store and we already know where its revenue would come from. It wouldn't be any different from Valve running Steam.
Google has nothing to do with the curation of any of these alternative stores, since they're owned and run by completely separate third-parties and Android as currently designed doesn't prevent a user from installing arbitrary app packages (apk's).
F-Droid is an open source alternative. Amazon has the Amazon Appstore. Samsung has the Galaxy Store. A more complete list here: https://en.m.wikipedia.org/wiki/List_of_Android_app_stores
Google has nothing to do with the curation of any of these alternative stores, since they're owned and run by completely separate third-parties and Android as currently designed doesn't prevent a user from installing arbitrary app packages (apk's).