To me, more than how the protagonist thinks and acts, the narrative framing and how other characters react to that character's meanness are the really important thing.
I'm perfectly happy to read a story where the protagonist is kind of mean or frequently thinks ill of those around them, or straight up does unpleasant things.Tyrion Lannister was one of my favourite characters in the Song of Ice and Fire books, and while he is comparatively well adjusted by standards of that series, he's frequently selfish, mean and manipulative.
But in Tyrion's case, it feels understandable because everyone is mean to him; he's always at a disadvantage, and while he gets his occasional victories, he just as often faces painful defeats and setbacks. He faces consequences and criticism for the bad things he does, or he does bad things because he's in a bad situation either by necessity or in a way the narrative never condones as being good. If the character is mean but people react believably to it and there are consequences, or at least if the narrator is describing it all with a bit of snark to clue us in that we're not meant to think this is actually good... yeah, sure, I'm okay with it!
The thing that will immediately put me off is when the protagonist does nothing but complain and lash out at people, and does bad things with the excuse that the world is bad to them and people are awful, but there's nothing in the way other characters treat them or the way their life is to act as evidence of this. All the other characters in the story are falling over themselves with a genuine desire to help them, falling in love with them because they're perceived as attractive and charming by everyone for some reason, and then the narrative doesn't seem to frame this critically at all, but writes about it as if this is all very sympathetic. A lot of people criticise Bella from Twilight for this reason; she's moved to a town she hates because it's...not hot like the desert and also there are... things to do and people... unlike the desert (wha-?), she automatically assumes everyone in this new town will be stupid and will make fun of her so she goes to school and just kind of mopes... and then somehow a bunch of people come and make friends with her like "WOW! You're from a desert!? That's SO interesting! Come and sit with us at lunch!!!!" and then the hottest guy at school is immediately obsessed with her... and Bella learns no lessons from this, and the narrative tone is never even slightly critical of how judgemental and negative she was, because the narrator seems to believe that Bella is better than everyone else; she's right to think other people are stupid and shallow compared to her, and people should like her.
A lot of teenagers (and some emotionally immature or narcissistic adults) find Bella sympathetic because they also think that they're right to think that they're uniquely smart, mature, deep and interesting, not like other girls, but it can be really cringe to read with critical thinking, because Bella is actually being really small-minded and mean, and in real life, that'd probably actually leave her isolated. It's an indulgent Fantasy where a person's misanthropic opinions and behaviour are justified because they really are better than other people and everyone should just deal with that.
So I guess for me, the line is "Does it feel like the writer is indulging in a Fantasy where they, through the protagonist, are "oppressed" and therefore justified in mean opinions and selfish behaviour while also simultaneously being showered with love and support, not really being even inconvenienced by the supposed "oppression" they say they face (the "bullies" never actually do much), and succeeding at everything?"