Yes. Most families we know in SF with elementary school age kids (and up) either put their kids in private school ($$$$!!), or moved to another city. SF has 72 elementary schools and a notorious lottery system that is unpredictable and confusing, so it can be difficult for families to get into schools they want. How the San Francisco School Lottery Works, And How It Doesn't | KQED
Apparently the SF school district is trying to revise the school assignment system in the next few years. Changes to Student Assignment for Elementary Schools | SFUSD
I think this particular school’s location in an expensive neighborhood isn’t necessarily relevant to the issues described. School assignment in SF is city wide so a student could end up anywhere.