As much as like F-Droid, it is only for open-source apps, isn't it? However, the majority of apps is still closed source. I would like to see an alternative android app store, which can also serve closed-source apps.

Another problem is a missing alternative to Google Mobile Services (GMS), which are installed on virtually all android devices. For example, if you need push notifications or in-app purchases you are dependent on them as well.

IMO, it is the widespread distribution of GMS (and a lack of alternatives thereof) that make google a quasi monopoly.