To me, the innovation of gmail was that it made a usable webpage, that worked the way a website should, and yet was still efficient and user-friendly. Part of this was done by copying the better aspects of the app world (keyboard shortcuts, eager fetching of messages, fast performance, etc). But it still fundamentally felt like an efficient, intuitive, information-delivery system (the web).
Copying the worst aspects of the app world (multiple scrollbars per page/screen, massively higher cognitive load due to all the scrolling, etc) is not an improvement.