Well it is odd in this modern age to discuss democracy and monarchy in the same sentence, wouldn't you think? What kind of power would you give her and expect to maintain any semblance of democracy? We Americans, for sure, would not want to deal with the poor dear with the authority of let's say King George III. We had enough of him and his General Cornwallis and the two Howe brothers in 1776 and don;t need to go through that again. 
Not in the least odd.
We have had that system in Britain for ages and ages, and mostly, it works pretty well. Much of its success or failure depends on the abilities and qualities of the person fulfilling the rôle, which is why there was so much anxiety - and still is - when people born and brough up outwith the confines of a life situation where pomp and circumstance and family responsibility and sacrifice are the basic raison d'ètre, come into the family through marriage. They don't have the background (and often not the culture) to cope, understand and take it on the chin when they have to do so. And if you think there is no sacrifice, ask yourself: how would you like to live a life where the spotlight is never off you and your relatives, where you have never enjoyed anonymity, and about the only publicity you are likely to get is that which comes in rabid piles when the press can find something to criticise you about, and over which to sell more trash news? And even then, the power of the person in the rôle is limited to ceremonial and representational functions.
And the value? The right person fosters a love of country, a rallying point of focus in emergencies such as war. The right person can - and does - exist as a person that can be looked upon as having the best of all desires and ambitions for his/her country at heart.
CONTINUITY is perhaps its greatest function, something that lasts as representative of one's nation, whereas politicians come and go, squabble and squander and generally fuck us all about in their insane power games and vanities.
No, it is not the same thing as being a pop star. They have no massive consequences other than wet, teenage panties or, alternatively as well as possibly in tandem, eyes.
As to your references to monarchs of a bygone age, pointless nonsense: in this day and age you have to look outwith Britain and the rest of the first world to find that kind of monarchy; you are much more likely to find that kind of presidency. Look around you.
Rob