I think there's some good argument that privilege is itself a cognitive bias - a complex one, that both builds on and encourages development of others.

It's not clear to me that privilege is a bias of its own, so much as aspects of privilege are examples of other biases, like availability bias.

I think the primary reason that attempts to discuss privilege don't go well is because the quality of most thought on privilege is, well, not very good. People who volunteer to speak on the topic generally have strong enough opinions that they can't help but moralize, which is something to resist whenever possible.