Well, all the killjoys can use logic and say that I'm using fallacies, until this thread runs out, but although other countries have monarchies of all sorts, ours is, I will assert(!), unique and is in a separate category of its own.
When people have made the effort to watch a particular occasion, and have possibly waited for hours to do so, do you see them frowning and looking grumpy? Are they moaning about the lack of democracy and the privilege etc? No, they are happy and smiling. Being there has made their day special. They'll remember and treasure the memory. This will absolutely NOT be because they feel humble, overwhelmed by, inferior to, the person of the monarchy, but because they feellllll a part of something solid and reliable. that is beyond price.
I worked in High Holborn at one time when I was about 20 and the Queen was due to pass by, so we all went out on to the pavement and saw the car drive slowly by. The picture is still in my mind. I have not been able since then to be personally at a big event in London for all sorts of reasons, but that's okay.