It's not just about grades, or being born on a wealthy suburb vs being born in a ghetto. It's all the other stuff, like people double checking that they've zipped their bag when a black person sits near them on the train. It's knowing that if you go for a job interview it's not enough to be qualified, you also have to have the right accent because there is an assumption that people who "sound black" are less intelligent.
None of this means that white people should hate themselves as individuals, or that a white person should pity a black person. "Checking your privilege" is more about trying to think before you speak/judge. Is the situation that person finds themself in because of their actions? Because of their race?
Another example. I have (slightly distant) relatives who are mixed race. They hate the police, and any interaction with them is almost always tinged with confrontation. Before I go wading in and say "Well if you didn't act so confrontational the police would be fine with you." I have to stop and think that I've never been shooed away from an area by the police because "I look like I might be causing trouble", I've never been "stop and searched" let alone had it happen three times in the same day while shopping round London.
All of these things are because of their race and over time have contributed to a relationship with the police that is tinged with distrust on both sides, before a word has even been spoken.
That doesn't mean that I feel pity for them, it just means that rather than tell them how to behave I check my privilege and understand that their behaviour is a product of their experience that I will never have because I'm lucky enough to be born white.
The result of "checking your privilege" in general should not be to feel pity, it should rather be your cue to either be supportive, or butt out.