I’ve thought of this idea too when we implemented the sidebar, before we even had an admin sidebar. It would be so nice to have an easy way to add links to the sidebar.
We went in a different direction with “custom sidebar sections”, which can be added using the 
button in the sidebar footer. This is not yet supported for the admin sidebar but we’ve talked about it.
What admin links would you add to such a custom sidebar, and which would you want available both on the main sidebar and the admin sidebar? And why?