Presumably double the density you also double the size of the school and double the teachers.
If you don’t, the problem isn’t the density.
If you don’t, the problem isn’t the density.
It's not that easy. If you doubled the density, maybe there is nowhere left to go for the school to expand, or the land is now so expensive there is no way for the town to afford expanding the school.
My town is an example. The middle school had two quite large empty lots on two sides.
Those lots have now multi-story apartment buildings in construction. Presumably those apartments will attract younger families who might have kids going to middle school. But if that school ever needs to expand, it is now boxed in by housing on all sides. Time will tell how it goes.
People rather have higher paying jobs, or quit because the job disappoints. I belong to the latter group. I quit teaching at secondary school (ages 15-18). After three different schools in three different areas, and having spoken with many colleagues and their experiences, I sadly had to conclude that broadly the work culture in secondary education (in this country, at least) is dramatically lacking professionalism. To a point where I think if my goal is to teach the next generation well enough, my time and energy is better spent elsewhere.
Only in medium density cities. Once you get to very high cost of living areas, finding and keeping teachers becomes extremely difficult because there's just not enough money to pay them enough for the cost of living. Many teachers either move to cheaper areas or change careers to high paying jobs.
A second problem is kids found out they can get everyone fired for claiming sexual harassment. Especially male teachers are scared. A brat utters a single sentence and you'll never work again as a teacher, and all your neighbours hate you.
Teaching today is a bad job.
A school was knocked down in my neighborhood to build a pricey apartment complex. The district and the developers made a bunch of promises about how current schools would be expanded and more resources would be available. Those were all lies, of course. Apartment complexes make money, public schools do not. Guess which one wins.