I’ve wondered the exact same thing for the exact same use case. Unfortunately the only answers I turned up was that it wasn’t possible currently, but I could be wrong. I’d love to be wrong in this case. But the way we handled it was just tacking the menu onto the end of the header bar in the layout like so:
This was the first pass, so the styling isn’t perfect, but it works the way we wanted it to.
Hope this helps and please let me know if you figure out how to get a nav tree into a widget!
Best of luck,