Furthermore, is the menu supposed to automatically reopen as well?
I think having a keybind for the menu is a MUST. This way we can open and close it as we see fit.
Reading the manual, Pages 66-67, would have help understanding the problem: there's no way from the SDK to programmatically open a custom toolbar menu from Simconnect, only Javascript can, but JS code is immediately killed the moment you click the toolbar icon to deactivate it so, once the toolbar icon closes, there's no way to bring it back programmatically, unless you reopen it yourself.
That's why the menu works that way, and sometimes (depending on context), it closes itself and sometimes it doesn't, depending what will be your next likely action, to save you from *always* click twice to open it ( if we left it always open ).
That's why there's no hotkey linked to it because, even if there was, it won't work unless the icon is active, so you'd still had to use the toolbar icon in many cases.
But even it it was possible to open a toolbar menu automatically, we can't have an hotkey assigned to it, because of ANOTHER problem in the SDK, which is the inability to prevent the simulator from getting an event reserved to an add-on, which was possible in FSX and P3D, is still fully documented as possible in the MSFS SDK, it just...doesn't work, the simulator will always execute the event no matter what, and if there's something assigned to the hotkey you chose, the simulator will do its thing together with GSX, so you would need to use an unused key but, if you later assign it to something, you must remember to fix the conflict with GSX, because there also no way to *know* a key used is...already used for something in the sim.