A diadem has nothing to do with a royal status. When you own one and are willing to wear it, it is perfectly free to do so. The real stuff however is expensive and, until the industralization and the development of modern economies, wealth was almost synonymous with aristocracy and royalty, the landowning upper-class. They were the ones who could afford big jewels and thanks to generations of precize distribution of inheritances, often still have these items.
A lady like the Argentinean-born Queen Máxima of the Netherlands happens to be married into a rich, reigning royal family and is confronted with a collection of awesome diamonds, pearls, sapphires, rubies, emeralds, whatever. She is also confronted with fantastic gold-, silver- and crystalware. With paintings, sculptures and other forms of artwork. With an enormous library and archives, even containing works, poems or music pieces written by the most famous of their time, specially for her husband's ancestors. It varies from a music piece written by the young Mozart for an Orange-Nassau Princess to a medieaeval handwritten and immensely beautiful illustrated Bible, from diamond-encrusted fans to complete treasure rooms holding gifts by inland princes from areas which once were colonies.
And Queen Máxima can do nothing. She can not sell these prizeless items. All of these have been placed in family foundations to preserve its best possible existence and for use by generations of Orange-Nassaus to come. What should she do? Leave all those fabulous items collecting dust? Not use the fantastic collection of ancient tableware, not use the carriages at their disposal, leave the diamond jewels in the cassettes? Rent the chairs from a catering company rather than using the hundreds of 19th C empire chairs from the own collection? Use the daily royal car for ceremonies and leave those fabulous carriages covered under a protective wrap inside the royal stables?
No, she decides to make good use of all this. Like Queen Margrethe II does. Like other Queens do (not all of them have such means at their disposal). It is also part of the "theatre of state" which is an important element of nation-binding and showing pride in one's background and history. It is exactly the same like the President in my country using carefully restored buildings, everything he (and the State of France) has at his disposal to show an almost royal court in a republican state. (The same happens in Italy, in Portugal, etc.). See
this picture. You essentially say that someone like Queen Letizia should not wear a diadem or follow a dresscode (like hats and globes). Look at these historic jewels of a Queen Máxima or a Queen Silvia. They can not sell it. They can not donate it. They can not break it up. They can only decide to wear it or to leave it in the cassettes and so will their successors...