LineageOS runs fine without Play Services. It's actually the default mode. It just limits your app choices. My needs are simple so I don't really mind, and I appreciate the improved battery life and speed.

It turns out many apps from the Play Store don't actually have a hard dependency on Play Services. Example: Slack works, except for notifications and (IIRC) receiving calls, which is good enough if you don't need to be reachable all the time.