How would you propose "splitting" a modern web app between two programs, one to retrieve its content and another to render it, without the sort of tight integration between these two programs that would be needed to again lead to the kind of "kitchen sink" situation that we have today??

IMO the problem is not the kitchen sink nature of web browsers, the problem is the kitchen sink nature of the web itself. It needs to be split into functionally separate parts that can be interfaced with different programs and workflows, instead of being a huge, integrated 'web' that is in turn parsed by a similarly huge, integrated program, the browser.