Seriiously, Catalina, did you even pay attention in history class?
This is why the American Revolution happened! For all the talk of "taxation without representation", it really just boiled down to us not being able to see the king wearing his crown.
Of course, realizing this, the founding fathers forbade any and all leaders in government from wearing crowns, as it would give them too much power, and they were supposed to be creating a democracy.
Granted, all sorts of other charismatic hats would pop up, but none as charismatic as a genuine crown, which on their own come with a positively astounding +5 charisma.
The entire point of the last panel is that the crown doesn't do anything other than be a crown. There is no magic to it whatsoever. I thought that would be clear in context, but I very quickly got questions and saw assumptions about what it "does". The answer is "nothing".
People have asked how Hanma could make a game like this, and this basically covers everything. Immortals can power artifacts that let people do things, so everything was fine so long as it was done through the game (hence why she didn't correct any "playtesting" issues as they went).
That on its own could have been enough, but individual immortal perception matters, and Hanma felt there needed to be prizes for this to truly count. These are not artifacts, but rather a scepter (that is a wand) and a crown created in a manner similar to Nanase's fairy dolls. It is safe to assume any future winners would get spells based on the version of the game they won.
Granted, this is non-canon, but there's sometimes continuity within non-canon... canon, so this wand and crown are just things Catalina can have at times in non-canon situations.