For about 2.5 years starting in 2015 I was a high school teacher at a Title I (low income area) school. There was a lot of anxiety and depression -- obviously not far outside trends around that. Very, very few kids were doing extracurriculars beyond maybe one sport, or a part-time job for a minority of the juniors and seniors. The (very) few kids who were oversubscribed the way you were didn't seem to have a rate of anxiety or depression greater than their peers, and the greatest incidence seemed to be in the lowest-achieving students.

Obviously this is anecdotal, and there can be multiple causes, but teaching gives you a lot of anecdotes and they don't seem to fit this narrative.