Please note what Apple is now calling the tint color, used to signify a tappable button, much like blue text on the web means it's a link. You think the web is "simply insane" now that most have abandoned underlines?
Hyperlink vs an actual button is not a good comparison. You don't have a mouse on iOS to hover over things to see if they're 'touchable' or not. Bad design is still bad design even if the cool kids are doing it.
There's no mouse to use to hover on mobile or tablet devices on the web, either. I haven't seen any push for change in how hyperlinks work on the web now that mobile/tablet is a common client.
Stretching the concept a little here, but design that hinges on being able to hover the mouse to detect possible interaction reminds me of those dreadful 90's web menus where you had to hover to see the title.
In your linked example, there are as many or more hyperlinks that don't appear as buttons. The button styling in those examples indicates "primary call to action" more than anything. A page that uses button styling for every single hyperlink would be a mess. The iOS 7 styling for "primary call to action" is to make the tinted text bold, rather than giving it a background color, though that's used in some different areas too under different context.
PS, a side note that these are 3rd party examples of iOS 7 UI. Certain things are going to be misused or overused etc.
When you have an app as well as a website, users generally prefer the app because it has a better experience. Even if your website is mobile optimized, users prefer native apps, at least on iOS. I suspect part of the reason is that iOS widgets (at least until iOS 6.x) are more intuitive than HTML. iOS 7 seems to eliminate this advantage, and Apple has given up much of the differentiation. iOS 7 looks closer to Windows Phone and Android, and even web, so it is unclear whether going forward, users will continue to prefer apps over websites by the same margin.
I'm with you on users generally preferring app versions. It's IMO a huge jump to say that it's because iOS 6 UI is intuitive. Users go through a lot of training, more than they may realize: friends showing their iPhones, ads on TV showing the iPhone being used, some UI concepts borrowed from the desktop paradigm people are already used to, and just that once a UI has been in the mainstream's attention for six years, people are going to have had a lot of exposure to it. After all that it's easy to say it feels intuitive. There are intuitive elements but don't overlook the amount of "training" that happens.
Responsiveness is another reason native apps tend to win out. iOS adds delays when tapping links in Mobile Safari, etc. Users also have expectations of consistent behavior for native apps, unlike the web which is its own diverse platform - iOS 7 resembles web UI in some ways, but don't forget the hundreds of pages of human interface guidelines it comes with.
Flat designs represented by these screenshots are a depressing step back in usability.