Hmm, do you have any examples of this? I'm curious as I've been running LineageOS without Play Services... seems to work quite well, but maybe I'm missing something?
The AOSP keyboard lacks functionality when compared to Google keyboard by default: there is no way to swipe on the keyboard to write words, and you cannot input emojis by writing their description.
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)