Perhaps because he perceives any kind of titles as divisive, non-inclusive, undemocratic, arrogant narcissistic, ridicules etc. Why elevate one person above another at all? He is a son of PEOPLE Princess, who would love to belong to the people, but he is unable to achieve this for himself without triggering of a big scandal. So he is doing his best to give it to Archie and his future siblings. Few small steps at the time...
That would be sort of ridiculous when the very foundation of monarchy is precisely elevating an arbitrary family chosen a long time ago above all others in the country and even giving that family the monopoly of the office of Head of State by order of primogeniture.
In other words, Harry cannot represent the monarchy and , at the same time, stand for any of what you said in your first paragraph as monarchy is exactly the opposite of all that. Modern Kings may have surrendered the actual governing of their kingdoms to democratically elected politicians with limited mandates, but monarchy itself will never be democratic or compatible with equality or meritocracy.
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