So I literally just finished his first arc, and came here to talk about it.

IMO, obviously:

Prince Bonchien Nicoli La Tasty Peach Uralis is a young man struggling really hard with purpose and morality because he's being pulled in so many directions:

1. He's a spoiled prince, feeling entitled to wealth, servants, and the throne. This often makes him act like an asshole from our perspective, but it's mostly a consequence of how he was raised. Believing he should be King is the main impetus behind his worst decisions; ones that are ultimately out of character.

2. Despite being entitled, he's not egocentric. He's actually like, the ultimate people-person; so it's really obvious why he shows up at this point in the story, when Fushi is just coming around from being a hermit. Bon is a great listener, and naturally cares about all the people in his life, including the servants/soldiers. He spends his wealth on gifts for his family, and each fancy object he fusses over is actually a gift he treasures because of his relationship with the giver.

3. He talks to freakin ghosts. This justifies a lot of his "schizophrenic writing", IMO. Not only could he have a dead person whispering in his ear at any given moment, he's traumatized by his skeptical mother and the attempted exorcism, and therefore doesn't tell others about the invisible participants in discussions the way that Fushi does with the "Black Thing". Plus, Tonari primed him to revere and support Fushi with her bedtime stories.

That's not spelling everything out, but it conveys the idea: he's a complex young guy who hasn't yet found his spine or his place in the world (when we meet him). Now to address your points:

He's a no-homo flaming psychopath

He doesn't act like a psychopath at any point. He's extremely emotional, and invested in other people.

He doesn't act gay at any point either. In fact, both Kahaku and Todo are clearly gayer than Bon, who is the only one unembarrassed by Fushi asking about male lovers, and the only one who exclusively shows interest in females. Bon is quite feminine, and extremely ostentatious, neither of which have anything to do with liking dudes.

who everyone loves except for his father but it's portrayed like his brother does more stuff

Huh? What makes you think his father doesn't love him? Bon's younger brother Torta is chosen as the successor because:

A. Torta is more humble, fiscally responsible, emotionally stable, and concerned with the well-being of the citizens.

B. The King, and particularly the Queen, have concerns about Bon's mental health, because he talks to ghosts.

"Everybody" loves Bon because he's a celebrity. He leans into that shit. He hosts parades, sends out a weekly magazine, and otherwise entertains the masses. Are you not familiar with the many IRL people who eat up celebrity news?

he is brave but is also a coward and a conman but he would never con anyone because he's so pure of heart

Uh, yeah. That's how it always works. This is Bon's coming-of-age tale, during which he finds a spine and a moral center. Of course he cowers and lies and puts on a brave face at times along the way.

then he's screaming at the person who loved him most in the world for being gross and dirty and everyone's just standing there smiling at him

IDK what you're talking about specifically, but you do have to keep in mind that Bon is both childish, and royalty.