You aren't talking (only) about policy change, you're talking about a financially untenable utopian solution that doesn't bother with the actual constraints of the problem. It reminds me of the old physics yarn "consider a spherical cow..."
Here are some constraints that I can come up with in a minute of thinking.
1.) Land costs. If you are building inside a major city land costs are huge, and you need acres of land. Outside of cities can be less expensive, but it's still a steep cost.
2.) Permitting/regulations/etc. I have no idea what it would cost to get the land you just built zoned to allow you to build a school, but I'm sure it isn't cheap or easy.
3.) Infrastructure costs. You have to build each of these new schools. The private boarding school that I went to[1] just built two new dorms after receiving a gift of $10m to house 98 students. They are probably really nice dorms, but it's not cheap to build. And that's just the dorms. You also need an academic center, library, gym, and possibly support buildings (like faculty housing, student centers, etc.). You also need to build roads, sewers, electricity, etc.
4.) You need to find and hire administrators and teachers. If you go with the chicago stats cited above then you'll need to pay each teacher ~$70k. One of the best things about private schools is the high teacher to student ratio. My biggest class was 14 students and most were in the 6-8 range. It's complicated to figure out the exact number of faculty off the top of my head, but you'll need at least 10 teachers. My high school had around 380 students and they have more than 60 faculty. So, for 100ish students you'd need 15ish faculty. That's $1m a year in funding. Chicago spends roughly $12k per student, which would cover faculty costs, but...
5.) You also have to pay administrators, janitors, lawn care, and a ton of other providers for service. Chicago administrators make roughly $120k a year. That difference has to come from somewhere.
6.) Oh, and you have to get accredited somehow so your students would be able to get into college.
There's probably plenty more constraints that we could come up with. But the biggest one is cost. Who is paying for this? A billionaire could do it, for sure. But it doesn't seem like people are upset enough about the current situation to pay more taxes to support the schools we do have. I'm not sure how you could convince people to pay to create a private school from scratch...especially if their kids aren't going to be the ones eligible to go.
It's going to take more than a minute to think about whether or not things are actual constraints. What you've done is list obstacles, which are not at all the same thing as constraints.
Look at your list. Can you think of alternative approaches that reduce or eliminate each item on the list?
