[UX] Move admin menu system tasks to their own dropdown?

I've always thought it kind of messy how admin_menu has placed common system tasks such as running cron and clearing caches under the home icon, without any sort of explanation for novice users.

Since we have added some other tasks to the right side of the admin bar, and are collapsing them into a "More admin tasks" dropdown on smaller screens, I'm wondering if we want to move the actual admin tasks to their own dropdown on that side.

The only tricky bit is that the dropdown menu items need to be moved to the left instead of the right:

And that can get a bit complex once all the extra items have been put in the top-level dropdown on smaller screens.

Here are the code changes I made to get that working like that: https://github.com/mikemccaffrey/backdrop/commit/c810735d5a1d1994b9aa801...

Do folks agree that it is weird that the system tasks are under the home icon? What do you think of this solution for moving them out of their into their own dropdown? Can you think of any ways to make this simpler?

One alternate solution might be to add the admin tasks to their own dropdown, but keep it in the main menu area, perhaps right before or after Reports. Is that a better idea? Should we bring that up in the menu reorganization discussion over in #1149?