A lot of this has do do with the area you live in. There are some that have more access to club sports at this age level. There are many places that do not. Especially for girls. Little league and t-ball is everywhere, but I have only seen softball teams for girls below the age of 12 in the larger cities.
You could be right. Most schools offer all types of sport here and you don't need to join a club outside of school. If I have to think what we had on offer it was ridiculous. Shooting,golf,surfing,rowing ,table tennis,tennis,swimming,water polo,rugby,cricket,hockey,squash,badminton,boxing (got stopped) and then athletics which included cross country. There were other sports like judo where an outside instructor came in.
Yeah, below the high school level the opportunities just aren't there unless you live in a more metro area. Around here it is mostly baseball and soccer and maybe some football for the kids middle school and below.