Not really...but sort of... :-) Read on...
Your example removed the reliance on the 'core-scaffold'. Rather, it seems you went about using it's individual components to recreate similar, but slightly different behaviour. In trying to implement your example into my existing site, it highlighted for me what was causing the appearance of multiple scrollbars. In my initial jsbin sample code, you'll notice I have specified core-pages height as '100vh' the top level polymer element (i.e. my-app-element). The second level polymer element (i.e. my-scaffold-page-element)...which it seems you've been focusing on fixing...loads within the aforementioned core-pages. Therefore, the nested scaffold element's maximum height is 100vh. Further down the chain, when the scaffold-element's main content area's height flows past its host's height limitation, it caused a secondary 'inner' scrollbar to appear, which has a different vertical extent than the original outer scrollbar...so trying to use the outer scrollbar alone doesn't effectively scroll the inner content to its entire vertical extent...forcing you to use the inner scrollbar as well to get the job done. Ugly to say the least.
Now that I know that is the case, one way to reduce the likelihood of an inner scrollbar appearing for the nested scaffold element's main content area is to change it's parent element's core-pages height to something much greater than 100vh (300 or 400vh perhaps?). Doing so solves the problem in a roundabout way. The outer scrollbar can now be used to scroll the entire vertical extent of the nested scaffold's contents without an inner scrollbar occurring. However, there is a catch...
It creates a side-effect I don't much like. Now, I cannot seem to 'fix' the drawer panel's header/main content toolbar (using your solution code) in place so they do not scroll with the content. I've tried setting different modes on core-header-panel and also tried using core-scroll-header-panel in place of core-header-panel along with it's 'fixed' attribute enabled...but no dice!
If this isn't a clear explanation, just let me know and I can update the jsbin with what I currently have.
Thanks again Rob,
Kelly