Play Integrity and Play Services are two different things.

Play Integrity is a remote attestation scheme by which apps can ask the OS to prove to a remote server that it is unmodified. It allows apps to refuse to run on devices with root or third-party ROMs.

Play Services is a set of libraries and APIs for things like network-based location, push notifications, and advertising. Nearly all Android phones include it, and users of third-party ROMs can add it at install time (but not later) with packages like MindTheGapps. There's an open source substitute called MicroG that allows most apps to run without it.