All of those activities are laid on in schools.
Your child can participate with no participation fee if you qualify for free or subsidized school meals. Schools provide transport to away games, meets, or matches. Basically you get your child to the school in the morning, and after that they are in the school's hands all day. I used to drop DD1 off at 5:30am for swimming practice in swimming season and the same the next season for water polo and didn't see her again until 10pm on meet nights, and 5:30pm on all the other evenings, when they just did before and after school practices.
Teachers are paid a decent stipend to coach sports, to lead a club (French club, astronomy, gardening, big buddies , wheel throwing, etc). DS' gf coaches a sport at the high school she teaches in and is paid 10k annually for this role.
My local schools provide endless opportunities in the areas of sport, art, music (all genres), performing arts, student publications, spoken word, debate, dance, social justice clubs, the list goes on and on and on.
The school made students do a one period study hall every day, when they could sit in a supervised room and do homework, study, or put their heads on their desks and nap.
The public elementary and middle schools offer similar opportunities, in sports, music, dance, theatre, clubs, etc