In what way is my opinion that the monarchy is archaic and institutionalises inequality based on 'gossip, hearsay, fake news, prejudice and jealousy?' 
The case against hereditary appointments in public life is straightforward. They are incompatible with democracy and meritocracy which, while not perfect, are the least bad ways to run countries.
I have little or no interest in the individual people filling the roles -- the problem with a hereditary monarchy is that by definition, you never know who you're going to get. Every family produces potentially mad or bad heirs. What then? There was, briefly, a British monarch with strong Nazi sympathies, an open racist who said that Aboriginal Australians were 'the closest thing to monkeys' he'd ever seen when on an Empire tour. If it wasn't for Wallis Simpson, he'd have been king for WWII.
Imagine the US changing their system, announcing Trump will stay in power until he dies, after which the crown passes to his son, King Donald junior? Would Americans stand for it? Or it's different because the British monarch has such limited powers? But how well-informed are people on the Queen's theoretically significant powers to dissolve parliament, declare war etc? Will they place the same unearned trust in King Charles, who, unlike his mother, makes his politics well known and has no issue with bending ministerial ears about them?
It appears to me that a large percentage of UK monarchists are so because they have a sentimental attachment to the queen which is unlikely to survive into Charles' reign, especially if he's long-lived enough to put the frocks-and-crowns excitement of William and Catherine's coronation on the long finger.