It's not mobile that has the limited version of the Internet - it's the PC
ben-evans.com
ben-evans.com
APIs are a solvable/solved problem - there should be no reason why web apps can't follow the same model for sharing contacts, accessing the camera etc.
Sensors? Depends which you mean. Web apps can get location data to a certain degree and can probably fall back to just asking in some scenarios.
What other sensors? I'm not aware of many killer social applications using my accelerometer or barometer.
There's a germ of an interesting idea here. More meat on it please.
Perhaps he should rewrite the article and replace the word "Internet" with "Facebook". Yes PC's are a cut down version of Facebook simply because the Facebook mobile app has probably asked for all sorts of permissions of your phone and you've "granted" those permissions because you had no choice. Is that a better internet for the user, or better for Facebook? We know the answer.
Except a PC knows who your friends are (email clients, Skype, other IM), the OS and browser know "where" you spend your time, PCs have a photos directory, and your browser might know what your credit card is.