Personally, I don't think it's helpful to pin racist or anti-racist as identities we can pin on ourselves or others, especially in an ambiguous situation where none of us can tell another person's intent. It's more useful to identify behaviours rather than make it about the person as a whole. So, yes, it isn't enough to just say one is anti-racist because it's not an identity one can just claim, but neither is "checking privilege" enough of an action either as that does the same amount of fuck all.
But then I'm a mixed (Mestizo) immigrant and not sure how much of myself would be part of the problem as the whole of me has the same opinions and actions. How I'm read by others differs widely from people doing gestures and doing that slooow ridiculous talk as they assume I can't speak English to people being surprised when a British accent doesn't come out when I speak - there is also plenty of other evidence that there isn't this bullshit idea of static privilege we can be constantly aware of.
Personally, I only care when someone hates me for how they think I look has the power to do something harmful with it, until then it's their problem and no skin off my nose. I have no desire to hyperanalyse other people's intent, make Whiter people question or have to show their anti-racist credentials or be afraid to correct me for fear it will look discriminatory, or pretend McIntosh's work from the eighties that as a previous poster said most of this is based on - where most of her 'invisible knapsack' has more to do with being a majority in a particular situation than race - isn't an oversimplified analysis that is only still around because too many lazy people keep dragging it back up over more modern works as it helped start a bunch of jargon that too many people think makes them somehow more ethical educated people. It doesn't, it just makes you as stuck in the past as a hypothetical someone still using a brick mobile.
Also, while reverse racism isn't a thing, it's only possible to say that people don't discriminate with White people if you treat White people as the default. If we move beyond the '80s and stop treating White people as default human and recognize people can discriminate positively and negatively towards members of a group, then it isn't difficult to see that people do discriminate when it comes to White people and that sometimes, yes, it's negative discrimination by those in power in favour of other people. Shocking, I know, but outside of jargon-laden parts of academia that recently got it's ass handed to it in the humanities paper scandals and lazy journalists riding along beside them, many of us work with nuance & mixed shades rather than black & white.