I have always found it quite odd that certain things i discuss end up on my facebook, and with recently reading that facebook still knows what apps you access when you arent on it makes me very uncomfortable using the social media app.
This brings me to why some people dont really like the idea of not having google on their phone. Whats so special about GMS that makes me or anyone else want it?
GMS is strictly a Google service. It's the backbone of all of their services as mentioned above and in the link I provided.
It is NOT required to run regular ol' stock Android. But, it's an extension of the additional services that Google provides.
Google makes the free Android OS available via AOSP (Android Open Source Project). It's the OS anyone can use and is the basis for many custom ROMs that are available like the Lineage OS line. Every android OEM uses it for their flavors and forks of the OS for their devices like Samsung, LG, Sony, OnePlus, Xiaomi, Huawei, and Amazon for the Fire Tablets and Fire TV devices.
Some of those OEMs sign agreements to also use Google Play Services (which includes the Google Mobile Services back-end). In order to use Google Play, you can't "fork" the OS. An OEM can customize the UI up to a point, but you can't change the underlying services. That's what Amazon does with their FireOS so they can provide their own Amazon services, and Google doesn't like that. So, Amazon isn't allowed to have any Google Services on their devices (they can be sideloaded after-the-fact, but that's another issue).
Being allowed to provide Google Play means an OEM must also provide all of, or at least access to, Google's apps - Play Store, Google Account app, Maps, Google+, Google App, Drive, all the Google goodies.
The problem is the tin-foil group that has issues with the data mining that Google does. Now, there ways to mitigate much of that, but there are also options for business users that can cut that out by paying for the services (paying for it means they don't need to make advertising money off of you, and privacy is tighter). But, it's a choice.
BTW, Facebook's data mining is a separate issue, and is why I disabled the app on my S9 and only use the browser to access it, along with the Blokada app to block the generated ads.
As to why Google in not on some Chinese made phones has to do with that privacy. Argue the politics any way you wish, but the Beijing Government does require access to information from any services that run within the country. Google won't give it to them, so Google is banned. However, a lot of the Chinese OEMs operate enough outside of China that they can have separate deals to include GMS on their phones. They just can't have it within the borders of China.
So, those phones get separate services added to work within those borders. And, of course, there are other "stores" that can be loaded to install the apps you want. They just have to comply with the Chinese Government data access requirements.
APIs are a different thing not strictly related to Google. APIs are a way for various apps to access information provided by any service. Can be Google, Facebook, LinkedIn, Uber, Instagram, Snapchat, whatever. Mapping information may be in an app window that pulls from Google Maps. There's an API that allows for that. An email app may have the ability let you share something on Facebook - an API on Facebook is what allows that.
That's, admittedly, still oversimplified. But, that basically covers the... um, basics. There are, however, deeper intricacies - maybe even a lot of exceptions - to all of the above.