The line drawing isn't strictly necessary -- It's more for debugging when you don't know what option you're on.
I do wonder why you want to do it like that rather than using objects, though. The video you linked doesn't show anything unusual.
That's because you're thinking of it like regular gui buttons (or a standard verb coin). The standard method relies on the mouse position being on top of a certain element on the coin or a certain region of the screen. Like a rectangular region. My menu works based on the particular angle, which means you don't have to have the mouse on the "button" to execute an action. It also means that it doesn't strictly require a rectangular area that regular buttons do.
Also when I have it working, I will add an advanced feature which means you won't have to even click too open and click to close the menu, but instead you'll just click-drag-release and it will do he required option without even showing the menu. (This means that eventually you'll know the options well enough you'll know what direction to drag the mouse in, which could theoretically speed up game play.)
It's alright if sludge doesn't do it, I'll keep searching. (AGS is too weird for me to use it.)