If I could respectfully add some nuanced points to this topic, weight stigma and the pathologizing of body sizes is another missing piece. The word “obesity” in and of itself contributes to weight stigma, medical bias towards thinness, and creates barriers that contribute to healthcare avoidance. If we are serious about addressing a systemic issue (in addition to poverty, mental health challenges, inequity in healthcare and access to food) we have to also recognize that language matters and we can all reframe our beliefs around body shape/size and health at the same time.
There’s also some discussion in my network (non-diet dietitians and Health At Every Size advocates) that “food desert” is a misnomer; because of the very issues you mention like racism, classism, built environments, etc. There has been a very deliberate attempt to limit equality and equity in certain areas and neighborhoods so food apartheid is sometimes used instead. Geographically speaking, yes, there are rural areas where there is simply no getting around the fact that grocery stores and farmers markets are few and far between, but in metro areas there is often an conscious, systemic reason why there is a lack of access to safe, affordable, nourishing foods.