03/04/2021 Meet Traversable Menu Module!
When we have a very complex multilevel menu and we need to make it extremely user-friendly the Traversable Menu approach could be the option.
I knew the JS library by Jim Keller (https://github.com/jimkeller/traversable_menu) that can provide such functionality out-of-the-box. I just had to wrap it into a Drupal module.
Now the module was released with the minimum number of settings. Please visit its page https://www.drupal.org/project/traversable_menu.