It's funny that in the OP the idea of seeing the royals as spiritual figures is supported by pointing at Egypt where they were seen as gods. While at the same time repeatedly pointing out that they do not have supreme power.
Egyptian pharaohs were more powerful than the kings of other regions because their office combined earthly kingship with connections to the gods and the immortal plane of existence. As pharaoh, these rulers not only controlled and owned all the land and resources of the kingdom, but they were also the religious heads of state, and thought to be the intermediaries between mortals and the gods. This was different than other kingdoms because they usually maintained more secular authority, and oversaw a priesthood separate from their individual personage.
That's what you get when you really smush together the spiritual and royalty. Incredible power.
So, what's the spirituality for a democratic nation or republic?
Well, there is no unity about that, at all. There is no national spirituality. You have the various monotheists: Anglicans (those that actually are religious), Catholics, Muslims, Jews...Then you have the really versatile Hinduism, now clashing with the less versatile Muslims right now in Leicester. After this you have all sorts of smaller groups. In history you go back to various paganisms, which were more or less indigenous but also came in incursions of AngloSaxons, Romans, Vikings. There's this undergird of paganisms. You have neo-spiritual heuristics, like stoicism (the stiff upper lipe, the cliche British stuffing down of emotions,w hich is a kind of implicit spirituality, even if it has not the slightest bit of spirit in it and can be quite secular.
So, what is the royal family that is a symbol for this mish mosh?
I don't think there can be one. And should a royal family be painted as this or decides itself to take on that role, it is misrepresenting its constituants. Perhaps once one could talk in generaliteis about the population of nationstates. But not anymore. For good and for ill.