FYI Even without GAPPs lineageOS contacts Google with IMHO way too much information, for example when you connect to a WiFi network a request to Google is made.
Example.com is for examples (so that if you copy & paste my code, you don't do weird things). I am not convinced that example.com would appreciate the worlds phones to use it at a whim..
(Plus, I'm not sure that's possible without controlling the domain. Not sure if the current ways to detect a capture portal are satisfied by a random 200 OK or actually _check the content_ of the reply. Which needs you to control said content)
I honestly don't see a good reason to do it although one of my friends pointed out that using TOR is probably one of the better ways to check for this since it won't rely on a specific server.
Firefox will make these requests as does Google Chrome and Chromium. NetworkManager will do them too and various GUI tools for network configuration will try it.
Otherwise Captive Portals would simply break your internet without notice.
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)
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.