"The fact that 37signals is unwilling to adapt to a changing market (i.e. demand for native apps) shows their archaic mindset and lack of innovation"
Mobile Development seems to be the next frontier in the evolution of the interwebs, and they could very well be betting on the wrong horse.
Web sites are cheaper to maintain upgrade, and add features to than native applications, which means that for the same price, you can provide more to your users. Think how much simpler it is to provide a web app that works on iPhone, Android, Blackberry and Symbian, compared with building it natively for each one.
They won't go away any more than PCs went away when we all moved from PCs back to the network the last time.