The problem is that the majority of apps written on Android these days depend on Google APIs or libraries, to the point that an Android without Play Services is almost useless. (Yes, this is from experience.)

Even Google's direct competitors require Google to use an Android. Skype and Outlook are great examples of apps you'd be amazed to discover will not run at all without Play Services.

There are fake GApps that I have tried on experimental devices and they fooled the apps enough to enable them to run. I haven't run these long-term or tried on more than 3 apps relevant to me to say if that would be a good approach.

I'd be a little iffy about security considerations once your plan is to rely on "fake apps".

I am not referring to mysterious binary blobs retrieved from torrents or such but open source projects such as https://github.com/thermatk/FakeGApps (http://repo.xposed.info/module/com.thermatk.android.xf.fakeg...).

People reading HN could, but the general public doesn't really have that option.

Does the general public care though?

Not at all. A lot of people who gather at places like these don't seem to realise that ordinary people have much more pressing problems in life than running a phone OS they can tinker with.

Does push notification still work without google services running?