The play services is a framework that many (and probably all "western") apps require because they use push notifications, maps, google ads and probably a lot of other google services. An android phone without those services installed isn't able to run said apps. And while it is possible to emulate the play services, I don't think this would work on a huge scale because they would still be accessing private google cloud apis and google could shut it down very fast. You have to remember, Apps don't send your notifications to your device directly, they send it to google, who then forwards your push notification to the device (if every app would regularly pull an notification server, your device would run out of battery way too fast).
The only reason google currently doesn't ban those emulations is that the % of users who use those is so low that it doesn't cause them any harm.
From a developer perspective, the Google Play Services are so damn convenient, and they work well.
When it comes to push notifications in particular, I don't know of an alternative which works quite as well as Firebase (if there is, please tell me!)
It seems like developing apps without the Google Play Services will take longer, and be more expensive. Maybe that's just psychological, but it's risky to decide not to use them, especially when you are time constained/on a budget, and you know how to use them etc.
There isn't. I have had to self host Gotify to send myself push notifications cos some proprietary apps do not have support to keep an always on connection to the server.
If you don't want to use google play, you can download the APK from any number of apk mirror sites and install the app yourself manually.