Not really knowing anything about the inner workings of the British royal family, I have no opinion about who is right and who is wrong. The only people whose opinion carries any weight are the members of that family.
But here's a generalization I've arrived at: Having too much money, too much luxury, too many priceless worldly goods, etc. is not good for people. It doesn't make them happier than the average person; it seems to be the opposite. This applies to all of the very, very rich, not just the royal family.
And before you say it's better than being poor, of course it is. Being poor is its own kind of hell.
I think the happiest kind of life is somewhere in the middle. Enough to feel comfortable and secure, to have choices as to where to live, what to eat, wear, how to amuse ourselves, etc. But not unlimited, excessive choices. I have heard of very few billionaire families who are very happy, or devoted to each other. Too much wealth seems to be a kind of poison.
But then again, why would the media concentrate on rich people who are happy and love one another? All we really know is what is presented to us. Those who worship the royal family have been taught to do so, probably all their lives. Those who hate them have been influenced by the tabloids and other media who hate them. Our opinions are formed by what is shown to us.