Spain? check guifi.net ;)
People had LAN Partys playing video games “offline” in the 90s… Setting up a network is easy, the difficulty comes from scaling up to many nodes, and spreading through the geography (e.g. if you were to use antennas for WLAN, they would need a mostly unobstructed vision) which in urban areas gets tricky.
But those “topology” issues can be flattened, e.g. you can always have a raspberry pi (or any device) acting as server in the corner of a neighborhood. A virtual bulletin board, emails, etc. all could be self-hosted locally there and then people could go grab a coffee and consume the local news just like in the middle ages, but with a screen, digital assets and some healthy amount of trolling :P
As other comments mentioned, Push Notifications, your main issue has workarounds. So, the answer would be Yes.
BUT Google Play Services is much more than that. This dependency can’t be worked around really and it’s one more way that Google establishes his stance in this oligopoly. I’m certain Apple has something similar if not more aggressive. This is the reason why the year of the Linux Phone is so far away.
Personally, I think Linux Phones will catch-up when their hardware allows for emulation of an android subsystem where we could sideload (illegally?) Google Play Services…