@PotatoZero First impressions are everything and unfortunately, our first impression of the Archduke is someone who ordered the execution of the MC on the spot even though she didn’t do anything wrong. At least that other princess he killed was resisting her capturers, but even then her death can be uncomfortable because her humiliation seemed unnecessary. Yes, he granted her a quick death out of mercy (I believe this to be true)... but not before cutting the front part of her dress solely to prove to himself that he was right about women. Many readers find MC to be very likeable and the Archduke is someone she is terrified of right now for very legitimate reasons. Perhaps they saw the possibility of the MC being humiliated and killed the same way.
Also, the setting of the empire holds many uncomfortable truths about medieval history such as slavery, torture, and hanging corpses on castle walls. The death/imprisonment of noncombatants can be very uncomfortable. The empire was not exactly portrayed in the best light to modern eyes and the Archduke is in a position of power within that same empire. The uncomfortableness of the empire could then be transferred over to the Archduke simply because he’s a strict follower of their rules.
Perhaps that’s what’s making readers dislike the Archduke. He’s too much of a stickler for rules that seem illogical to many modern readers. Case in point, sentencing Howl to death for one single mistake. We do get a look inside his head portraying him in a better light, but... he’s still going through with it. Yes, he chose not to execute the MC... but only because his vice captain suggested that she was worth more alive than dead. It was also the vice captain who stopped that red haired guy from raping the MC, not the Archduke. I don’t think he was even in the room when that happened. Either way, red haired guy did not appear to have been punished, so readers may wonder if the Archduke was enabling his behaviour. They both seem to share the same mindset about noble ladies, after all.
Perhaps the biggest problem readers have with this guy is that he’s too decisive and pragmatic in his approach, which are good qualities to have as a military leader in life and death situations. Problem is, that mindset has been a threat to the very sympathetic MC on more than one occassion. It’s also the sort of mindset that leads to conclusion jumping. On the surface, he seems narrowminded if not for the fact that he does listen to others like the VC and the MC, but that says more about them than it does about him. They had to speak up right away because he had already made a lethal decision right there on the spot (or was going to if they stayed silent for more than ten seconds). He’s also a bit too prone to resorting to the kill option against noncombatants. That sort of uninformed quick decision-making can rub readers the wrong way (he didn’t even know Beatrice was Gilbert’s wife when he ordered her to be executed!).
Tldr: Archduke isn’t hated because he’s not worshipping MC. He’s hated because he’s impulsive and an active threat against a very likeable and sympathetic MC.