I was teaching in a low level public high school. To be fair, the entrance exam system is complicated and I don't fully understand all of the details, but I'll try to elaborate some more.

Essentially you are allowed to apply for 1 public and 1 private high school. Unlike most places in the west, public high schools still cost money in Japan. It's quite expensive: something like $1000 per student per month. However, the government helps you out if you are low income, or if you have multiple children -- so essentially it's affordable, though expensive for everyone. Private high schools can have higher prices, but not always. So it's not necessarily the case that a private high school is much more expensive than a public high school, especially if you are middle class and only have 1 child.

I think the main reason for the 1 choice only was that it used to be that students would write many exams and then pick the best high school they qualified for. What the new system does is make it so you have to strategise a bit: you can apply for a higher level private school (maybe aiming for a scholarship) while applying for a lower level public school. The main result is that students tend to lower their sights to ensure that they get in.

However, there are caveats: What happens if a student doesn't pass either of their exams? What happens if the student doesn't pass their public exam, but can't afford the private school? So there are extra exams. Essentially, the schools pick the students that they want, but leave some spaces open. Then they do another exam and fill up the remaining spaces. So there is always a possibility that you will get accepted in the second round. Access to education at the high school level is assured, so if you totally screw things up, you will still go to school -- but you may go to some random school.

Anyway, it really is the case that students as low level high schools don't study, generally! I don't know how old your child is. You said teenager and given your story about required sports club, that's almost certainly junior high school. Sometimes if you live in a very rural area with only a small school, then you can also be stuck with what you are stuck with. But if your child enrolls in a school that is aiming the students towards trades, or factory work, the academic expectations are practically zero. If you enroll in a school with the goal of going to university, then the expectations a very high. Keep in mind that the vast majority of Japanese students do not go to university -- it's very different than the west. Usually they will go to a kind of junior college (like 2 year dental assistant program), or a trade school, or they will go straight to work.

In my school (which had about 750 students when I taught there), 50% went straight on to work. Normally the number of students going on to university I could count on my fingers (and sometimes only on one hand). In my area there are 4 high schools. 3 of them are similar to the one I taught in. Only 1 has students who are expected to go on to university. This is totally normal here.

The vast majority of high school students in Japan have an easy time of it. You child may not, especially if you expect them to go on to further education.