Does anyone know if these game services actually get your app more downloads? They seem to be useful for leaderboard hosting, but nobody even uses Google Plus.
I'm not sure where you get this from, but you couldn't be more wrong. Everyone on native Android uses G+. Having a Google-ID to use for your phone basically makes you an automatic citizen of G+, because elements of G+ drive so much of the backend. Hangouts? Is a part of G+. Do you backup your photos via Google's automatic services? You're using G+ (now split out to Photos). In a sense, G+ is the glue which ties together all of Google's services.
In another sense, G+ doesn't really exist any more. Hangouts for business no longer requires a G+ id. Photos has just been split out into a stand-alone App. Streams (the feeds aspect), is in the process of being split out. I suspect that in future, using Google services probably won't require a G+ Id (following the lead of Hangouts for business). But you'll still have an ID with Google which is really all that they need - and which will tie together all of your Google services.
Google Play games is just one of those services, and it has little to nothing to do with G+, other than using your G+/Google ID.
I don't know much about Apple Game Center other than it is similar to Google's. Do most users even bother to sign in to these things when the game is in play? Games such as crossy road, cut the rope, etc. use them for leaderboards, but I can't figure otu if it's worth the trouble. Thanks.
The question is rather, why would you waste time implementing this for yourself, when you can get the entire thing - including backend services - essentially for free?