@Sébastien what do you think? Is this working as intended (touch causes the overflow to scroll, mouse causes a drag and drag of the icon)? If not, what should happen?

Then reassign to me :-) Thank you!

This is working as intended, in both cases it triggers the the scroll action as expected. I do not think overflowing the overflow is something I'd like to see.

Quick question, is a vertical mouse scroll wheel also horizontally scrolls the overflow? If not that's an issue. 
Another thing to consider might be to add scrolling affordances on the left and right.

cc'ing Kejia and assigning to Gary for further investigation and longer term planning there. This will be come more of a thing as we get potentially smaller devices.
@Gary, depending on the answer to my question, some UI input will be needed.

Ok, I'm going to close this bug report because the originally reported issues, as described in the original description, are working as intended. Please file separate bugs for other things that don't work as you expect. Thanks!

Something to keep in mind (which is not a bug deserving to be reported independently):

Now that we have three different environments (Chrome OS browser, Android, and currently-beta desktop Linux), a single overflow shelf no longer suffices. Either additional overflow shelves or a more hierarchical menu structure will be needed sooner or later. Since some of the same applications likely will exist in each environment (e.g., web browsers), a more hierarchical menu structure will work better and cause less confusion than merely additional overflow.