A rather long time ago Microsoft introduced the Edge web widget, which had limited functionality and customization.
It seems while we have not been looking the feature has seen sudden improvement, with a quite powerful floating web widget now available in Edge Canary.
The widget can be activated in Edge Settings from where it can also be customized with various modules, including websites of your own choosing (as long as they are responsive and fit into the narrow profile).
It now offers an always-on-top floating button that floats above other apps, allowing you to access to Edge bar instantly.
It can also be resized in various ways and docked to the edges of your screen. You can even have it start up when you restart your PC.
Check out a video walkthrough of its features by Brentech below: