1) There are many more web users than there are mobile users (author claims 2 Billion vs 50 Million)
2) There are some browser apps (Angry birds and NY Times cited) which give you 'offline' access.
3)The HTML5 in the browser will become a unifying API to local functions which will ease development.
So lets take these claims one by one and see how they hold up.
Market Size - ok there are a lot more web browsers than mobile users. And yet if you look at growth rates (especially if you count 'tablets') the number of users with 'mobile' capability is growing faster than pure web browser seats. If you look at user capability the 'mobile' user may want to do things like 'check in' (location based) or navigate which the generic browser user doesn't do. Finally if you consider the underlying UX tools its just as painful to make an App which works well on 1920 x 1080 screen with mouse/keyboard as it does on an 800 x 400 screen with just touch points.
Browser Apps with offline access - This moves the browser closer to hosting the 'OS experience' of the device away from the underlying OS. Google would love to do that since they would realize Sun Microsystem's dream of being able to ship a browser which was the OS can cut out the OS provider (either Microsoft or Apple in this case). However adding this layer not only impacts performance, it also impacts compability since browsers with the same name and version work differently from platform to platform. Not so with mobile apps.
HTML5 will develop access to native APIs - I read this, and I imagine all of the zero day exploits that will come of it, if it comes to pass. You take two engineering teams in two separate companies, one providing an OS API and the other providing access to it, and you get Flash all over again. Both development teams come from the problem from different directions and create unexpected problems for the other.
Bottom line for me is that I'm not convinced the browser can kill anything except other browsers. That being said, I also think there are big opportunities in mobile / tablet device space for innovations in the operating system, the UX frameworks, and the distribution mechanisms.
Disclaimer though is that I was one of the Java guys and I had completely bought into the notion of 'write once, run anywhere'. Making that true turned out to be impossible for a number of reasons, not the least of which that the underlying OS providers were hostile to the idea. So creating an HTML5 uber-alles experience for developing apps seems optimistic to me.