Before one can rightfully state that “you do or don’t need the play store” one has to define “you” and “need” and differentiate between “play store” and “play services”.
- “You” can be a person who is used to having all the fancy apps that come from the play store or a person who uses apps that do the same but come from different - less fancy - sources.
- The feeling of “need”-ing a specific app can be caused by not knowing any alternatives that offer the same feature or by being restricted by the environment (peer pressure, workplace, …) to use a specific app.
- Needing the “play store” is only true for apps that you can’t download from anywhere else (nowadays I don’t believe there are any), while “needing play services” means that the app refuses (for no apparent reason) to work if it doesn’t detect play services on the phone. This can usually be circumvented by installing the free microg services. (AFAIK they do integrate some proprietary Google services but do so in a controlled environment so they can’t communicate any personal data to Google’s servers)
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who is doing the push-service for you?
That could be microg. If interested check 5.5 in this guide.