I do. The only Google service I actually use anymore personally is GMail but that simply forwards to my iCloud account. It's easier and more efficient on battery to just check one mail account. I also ditched Google Calendar / Contacts for iCloud because they work almost exactly the same. Again easier to just use/sync one service IMO. I was never a big Google Docs user to start with but now I use DropBox and native apps. Again I only setup my DropBox once and it's good to go. I can use different (and more powerful) apps as needed. Sometimes Numbers on iOS is fine for quick data entry but for building out a big spreadsheet I'm going to most likely use MS Office on a desktop. I guess you could do this through Google Drive now but I already have DropBox setup so why bother? I use FaceBook/Twitter instead of Google+ which are both first class citizens on iOS/OSX now. FaceBook chat has replaced GTalk just because everyone I know uses it. A lot of this goes back to Google's period where they did a very bad job supporting iOS. The path of least resistance was just to use something else and now I have no need to switch back. From what I've heard their iOS apps have improved greatly but I'm happy with what I'm using.
Google Drive, OTOH, works on Windows/Mac OSX/Android/iOS (and probably WP7, Windows 8, Blackberry, etc, but I haven't done my homework). Dropbox is another option.
Personally, I would never use iCloud. It's not because of trust, it's because of portability. What if I have an iPhone, then decide to buy a Samsung phone? I can't use iCloud on that device (AFAIK), so I'd have to transfer everything manually. What if I change from Android to Windows 8? AFAIK, Google Drive will still work.
If you're so bent on trust, then use Dropbox. It's much more portable and they don't really care about your data, just your monthly investment.
He's willing to use the iOS ecosystem and it satisfies his requirements and it's convenient for him.
Apple has technical limitations in place to prevent alternatives from being as good as the default. A simple example is setting the default browser/email application. I'm not sure if this is still true, but Apple had a policy preventing 3rd party browsers from being anything more than a skin over Safari.
Since Apple is in complete control of the App Store and there aren't any 3rd party app stores, Apple has no incentive to allow alternatives to it's default applications.
This feature has been around for either the entirety of or damn near the entirety of iOS programming.
Power users certainly aren't the majority of the market, but they do tend to steer things. There are, of course, many barriers that keep users from switching platforms, but every little bit helps.
I do like your idea, and agree Apple should allow this. They should also allow changing the default applications, and allow better access to internet APIs, so Mozilla can actually make Firefox for iOS. But that aside, what you're complaining about isn't something Google does either