You know, I find the ‘Alicorn Tetrarchy’ thing weird because Twilight is implied to not actually rule anything, with her status as royalty being a title, rather than an actual position of rulership. She clearly had no ruling power in Princess Twilight Sparkle’s 1 and 2, and even after getting her castle, Mayor Mare is still Ponyville’s main governing pony, and Twilight herself doesn’t seem to have any hoof in how Ponyville runs (beyond occasionally organizing things).
 
As far as Twilight is concerned, her ascended societal status mostly seems to be a thing of responsibility in some situations, such as overseeing and organizing major events. Roles of high honor. Otherwise, her Princesshood just basically seems to be social status rather than an actual part of the ruling body of Equestria.