Maybe achievable with a theme module? I’ve seen similar tricks: Adding Unsolved button to top menu using custom HTML
If not then you should probably roll your own extension or fork of this plugin, because hiding the category drop down is highly non-standard.
So basically the same as the “move favorited categories to top of categories list” stretch goal right?
That’s some prime real estate… I can support it as a stretch goal. I think it should be behind a setting, because this is more in your face than any of the other UI we’re adding. It adds a choice + unknown to what is currently a single option.