You can provide Google OAuth sign in without using any of Google's other services. Your OAuth client would just have an empty scope. You should be able to get the basic user info and email (so you can figure out who is who by something other than full name) without enabling any services as well.
You will still need a way to flag that you are you in your app (probably just hardcode your email in your app, then check it against who authenticates with OAuth would be the easiest)