Is there method to manage the visited state of the default menu?

1)  Articulate doesn't provide a way to adjust when the tick mark appears on the built-in Menu. A slide is marked when its timeline starts. (That behavior could probably be altered with JavaScript, but I can't help with that.)

2)  You can track completion of any slide using a variable. For example, create a T/F variable with the default value of False. Add a trigger that changes the variable to True when the slide/interaction is complete.

• The "when" in the trigger might be when the timeline ends, when designated objects are in a Visited state, when the user reaches the final layer in a Carousel activity, or whatever means the slide is completed. 
• A variable's value stays the same until a trigger changes it. And the value can be accessed from any slide. That means you could use the value of a slide's T/F variable to control what happens elsewhere in the course.

FYI: The interactive slides in your Sample file have a trigger to disable the Next button when the timeline starts.

• That trigger will run every time the user visits the slide, which means the Next button will be disabled even if they already completed the interaction on a previous visit. 
• To prevent that from happening, the trigger needs some kind of condition(s). 
• This post has a demo for different ways to control the Next button: TIP: Controlling the NEXT Button 101 - Articulate Storyline Discussions - E-Learning Heroes

It’s worth the effort to learn about variables and trigger conditions, because they provide the real power in Storyline. Here’s more info:

• https://community.articulate.com/discussions/articulate-storyline/the-value-of-variables
• https://community.articulate.com/series/articulate-storyline-360/articles/articulate-storyline-360-user-guide-how-to-work-with-triggers#conditions