Adversity is usually college, or at least after age 17, but some people do use struggles from when they were younger. I was thinking, too, of the assault which happened in college and would not factor into a disadvantage statement but which could work for overcoming adversity.
If you come from a high SES family, then you are going to be working against the trope that an applicant from a high SES is not "disadvantaged". If you were homeless and couch surfing for months, then maybe you could get away with it, particularly if you were not provided with opportunities for school field trips, after school activities, hobbies, music/dance lessons, etc.
Are you at all interested in women's health/medicine? If so, you could work some of this material into a personal statement about your own experiences and your interest in serving women, including those who are marginalized by cultural norms and expectations. There is more than one way to get your story heard. The trick is to do it without engendering prejudices against you for self-identifying in a way that seems not to fit the circumstances.