Update super-sidebar styles and add overlay

What does this MR do and why?

Address the feedback from #392444 (closed):

• Add tests for the state manager
• nitpick: This could be simplified to not call the function twice
• thought: Is there a plan to add an clickable overlay to hide the sidebar when on mobile? This is the behaviour of the existing sidebar
• suggestion (non-blocking): Should this be super-sidebar-width
• suggestion (non-blocking): WDYT of moving these two utilities into the definition of the super-sidebar class

Screenshots or screen recordings

Super sidebar mobile overlay

Light	Dark

How to set up and validate locally

• Enable the feature flag

Feature.enable(:super_sidebar_nav)

• Enable the "New navigation" setting toggle from the user dropdown

MR acceptance checklist

This checklist encourages us to confirm any changes have been analyzed to reduce risks in quality, performance, reliability, security, and maintainability.

• I have evaluated the MR acceptance checklist for this MR.

Related to #392444 (closed)