When I'm focused on a project, there's a decent chance it is a sequential project, and so a next action project mode view with the sidebar collapsed offers minimal distraction. If it is largely parallel in nature, or I'm focusing on a handful of projects (the most common scenario of the three for me), a context mode view grouped by context or possibly due is going to be the most convenient. In either case, the sidebar holds no useful information that isn't also found in the outline panel, so ignoring it comes naturally, and I may close it just to reclaim the real estate.
I don't understand why you regard the closed sidebar, collapsed context groups approach as "icky":
The former seems cleaner to me if I'm trying to avoid distraction; no sidebar to click on, no duplication of context names, no "No Context" bin that doesn't have any relevance (to be consistent, I think you should be asking that it also not be shown if all the actions have contexts). Depending on the mix of items across the contexts, I might still go with that project mode view.
For what it is worth, I don't see anything intrinsically wrong with the behavior you propose, but I think you can already do better (given your stated reasons) with the product as it is.