> The most common chemical sunscreen ingredients cause cancer themselves
Both of these statements are out-of-date.
Recent studies show two different histological subtypes of melanoma for young adults vs older adults, each following a different common mode of presentation [1]. Melanoma in younger adults tends to present on the long limbs and is thought to have different contributing factors.
Also "common chemical sunscreen ingredients cause cancer" is a common FUD trope. Sunscreen reduces cancer mortality, full stop. Each person's circumstances vary based on their daily habits, geography, and personal/familial history.