I’ve proposed internally that we have a button on all site pages to add the page you are on to the sidebar. I’d love to see that happen, to make it super easy to decide which pages you want to have at your fingertips.
I don’t think it is a good idea to mix notifications with add to sidebar but a new button somewhere in the UI would make sense to me.
I agree with distinguishing notifications and lists/links. The notification hamburger imho has two categories belonging to the sidebar — bookmarks and user menu. These are not notifications but links to other parts of the site.
Bookmarks are a little bit of a special case, because while they are a list, they also produce notifications when reminders are set. We tested bookmarks in the sidebar internally and the consensus was that people really missed them in the user menu.