After a short escapade in the land of tiling window managers, I have managed to get Gnome to resemble the Unity-like behavior I described in the question.
They key is to use jumpapp with the -t switch and specify the title of each webapp in order to switch to it exclusively. Now I I can use a keyboard shortcut to run or raise / launch or switch to each webapp and webapps also show up as separate icons in the gnome dock. Webapps still group with the browser window under the same icon for the Alt + Tab switcher, but I can get around that using the jumpap hotkeys instead of Alt + Tab for those windows.
Update 2017-10-28 I have since switched to using the tiling window manager i3, which I enjoy greatly. jumpapp is still a cornerstone in my workflow and it works well together with the i3 desktop, although I do not use dedicated web apps any longer.