I've been running LineageOS 14.1 on a Sony Xperia tablet for several months without Google Play Services or MicroG and am not noticing any missing basic functionality. I do tend towards open source apps and away from Google stuff but I'm by no means strict about it. I use Yalp and F-Droid to install apps.
Some of the apps I have installed (eg. SnapSeed) complain about missing Google Play Services and say they won't function but I ignore the warning and they seem to operate correctly... if there's some missing functionality I've yet to encounter it, I'm happily editing and exporting photos.
I'm certain that some apps would fail to function without Google Play Services, just that I don't happen to have fallen foul of any of them. My usage is probably not hugely standard but I use modern browsers (Chromium/FF), video/audio players (VLC/Black Player/Spotify), K9 for email, browse and edit RAW photos... and jump into Debian Stretch via Linux Deploy + Xserver XSDL where I often do web dev work with tmux/vim and proper desktop Chrome/FF including Developer Tools. I'm using a Bluetooth keyboard and speakers... can't really think what basic things I might be missing?
EDIT: (this is just the standard install of LineageOS btw - which doesn't come with Google Play Services, you have to install them separately - I just skipped that step)
How's that work for you? I've tried it but it seemed a little rough.
Play services is responsible for connectivity with Google's servers which are used to distribute notifications to the device.
Play services is probably also responsible for backing up the App's data to Google servers (not sure about this one)