We had a web for applications. It was called telnet. AOL, Prodigy, and others all had actually-pretty-sophisticated remote application deployment systems with vector graphics and fancy input processing. In the early days of the web, it was also common to write line-of-business apps using tools like Visual Basic, Hypercard, and even Excel (some things never change). The web won over all of them because, as history tells us, there are fitness advantages (e.g. platform agnosticism, easy introspection, and gradual enrichment) to evolving application capabilities into a widget toolkit that you just don't get if you try to make the optimal widget toolkit up front.

If Tim Berners-Lee had made the web for apps and not documents, history would have been no different. Somebody besides Tim would have made a remote document system and we'd all be using that instead of something called the web.