I was thinking about this based on my school days. I'm not sure that there are any truly "accessible" sports, inasmuch as there are no sports which people don't benefit from having more money.
At my (comprehensive junior and Grammar secondary) schools, the gymnastics squad was only open to those who'd previously done gymnastics - which meant private gymnastic clubs/classes were essential. I lived close to a swimming pool, and my parents would pay for me to join all the swimming holiday clubs, and I wasn't a bad swimmer (I'm still better-than-average, although a shoulder injury put paid to my swimming days recently), but they couldn't afford the early morning before-school private tuition that the "good" swimmers at school got. Even the girls who were in the all-England under-sixteens (or whatever) hockey and netball teams were coincidentally from reasonably well-off families, while my friends who were couch-surfing between divorced parents' flats were always left on the sidelines...
It's always going to be the case that no matter how much love, encouragement and support you give a child outside of school, they'll never do as well as the child who has love, encouragement, support and money.