Fencing for example, is usually clustered around external clubs. Very few high schools will have fencing teams, and in a lot of cities even the high schools that do have fencing teams will be kind of a joke compared to the club teams.
Of the others, there are either homeschool alternatives that are explicitly secular or at least not overtly religious, or there are competitive clubs. All the schools have track & field, but there is a large homeschool league. And the district has a few schools with pools and a few more with swim teams that practice at the city pools, but the local swim club is the one turning out the Olympians – but even then, it also seems to have plenty of offerings for kids who won't set a world butterfly record. Football, I imagine, is just so popular that the private/public schools take all the players.
Sports might be the challenge. Many US states have athletic associations that handle most K-12 sports, and they require enrollment in an accredited member school. I am aware of several homeschool specific athletic associations in my area, but all are targeted towards religious homeschoolers. Not certain what secular alternatives would exist, but soccer is very popular & there are plenty of competitive academies that operate outside the school ecosystem.