> general purpose WebGIS app
> QGIS is often too heavy or inaccessible
The reasoning is the contradiction here. While most GIS work is conceptually simple, there's so many things people want to do - any new GIS program will either feel incomplete or end up getting incredibly bloated and complex (per QGIS). Try dealing with raster+vector, different file formats, different databases, projection systems, different web service standards, tiling formats, 2d+3d+temporal visualisation, proprietary libraries for particular analysis, etc. etc.
Not to say QGIS can't be improved or a better webGIS wouldn't be helpful, but its a tough task to do so many things even if it all seems conceptually simple.
Lets not forget QGIS itself was basically a clean-slate approach which started much less able than previous open source GIS (Grass).