Thanks to Kari for bringing this issue to my attention. I just want to share our experiences and attempts at resolution. If any of you have any suggestions, please share them in the comments below.
At this point, almost all of you are using Google Calendar to organize your days, weeks and months. I’ve said before that I think it’s a great application. I prefer it over iCal (or Apple's iCloud Calendar, to be exact), just as I prefer Gmail to Apple's Mail.app. Google Calendar is just as intuitive, but I find sharing events, inviting participants, and the overall control of Google Calendar to be more comprehensive than its competitors. So if you’re using it (and you should be) then good on you!
I should also say that I think the Gmail, Google Drive, Docs and Hangout apps for iOS (iPhones and iPads) are really well developed. So good on Google, too.
Google Calendar was released as a mobile app for iOS in March of this year (2015). And I have to say it’s a big disappointment, as far as I'm concerned. Does it look great? Sure! It’s far more navigable than the native iOS Calendar app. Is it fast? I guess so. It seems to sync up fairly neatly, when I alter my calendar via Chrome. Does it do much of anything beyond that? No. In fact, it kinda stinks that way.
The specific problem that many people run into with Google Calendar for iOS is when trying to invite someone to an event. In my case, and in the case of others, the contacts simply weren’t available to invite. At first, I thought this was a simple issue of having the proper accounts set up on my phone. Well, it was and it wasn’t.
You would think that once I imported my school contacts into my mobile Gmail app, everything should be synced up properly. But that doesn’t happen.
You would think that adding a new “Google” account in your Settings > Mail, Contacts, Calendars on your phone. But that won’t work either.
You would think that the old way might still work. At one time the solution would have been to add a new “Exchange” (Exchange ActiveSync) account, but evidently that’s not the solution anymore. Google doesn't use the exchange platform.
It would appear that the best way to import your contacts so that Google Calendar will recognize them is to add a CarDav account to your phone. See this article on how to do that.
OK, once you’ve done that, you can re-launch the “Contacts” app on your phone and it should begin syncing. You will not be notified when this process is complete. And it’s likely you will have duplicate (or more) entries for each of your contacts. One for your iCloud contacts which were already there, one for your Gmail contacts, one for your new CarDav/Google contacts, and probably more. So you’ll need a merging/cleanup app for that. I used two: "Smart Merge" and "Cleanup Dupes". Both of these apps work well—just make sure you've backed up your contacts before you go messing around in there!
OK, now once you’ve done that, you probably have a 50-75% chance that Google Calendar for iPhone now recognizes your contacts. In my case, many of my school contacts show up when I want to invite them to a meeting, but many still do not. Including my wife. Hmm… or she may have set it up that way.
Anyway, my suggestion is simply: don’t use it. It’s still way too buggy. Or if you just have to use Google Calendar for iPhone, use it as a “View Only” app; don’t expect to manage your calendars this way. A really good alternative that Robert Appino pointed out to me is Sunrise. I found I was able to sync this app up with all of my contacts from various accounts and it took no time at all. It's free, it's intuitive, and it does exactly what it says it will.