Almost (probably everyone) has biases. That is nothing new, and I would think is uncontroversial. Allowing your biases to dictate your behavior in an uncritical way can be damaging, either for yourself or others. That I also think should be uncontroversial.

People who grew up in a given culture tend to have shared biases. Some of those will be useful, some will be harmful. This is not to say they that every member of that culture shares those biases. My guess is that what you're talking about is the tendency to assume that a particular member of some culture has a bias that is common in their culture as a whole (for example, to use a US-specific example, if I assume that any Southerner I meet is biased against socialistic ideas). This is clearly not always going to be accurate, but may be an assumption made for safety's sake when you're in a vulnerable population and you know that those biases can be damaging to you (if I'm secretly a communist living in the South, it may be better to hold that in on average to avoid problems).

I think the reason these conversations may seem "targeted" at well-off white people in the current cultural context isn't because other groups don't have biases (they do!), or that every white person holds a given bias (they don't!), but because well-off white people on average hold more power, and therefore their biases are as a consequence more likely to cause harm.

And also, sure, I'm sure there are people who go overboard with all of this, but that is true of literally any position. Letting the extremists define the discourse isn't going to help anything.