One of my kids went to a magnet school for performing arts. They’re public schools, and they don’t cost extra, but you do have to apply for them if you don’t live in the immediate area and there’s an element of competition to get in.
The idea behind magnet schools is that you can improve the overall academic performance of a school if you increase the diversity of the economic and academic backgrounds of the students. Schools in the US are tied to a geographic area, which in turn is generally correlated with income - poor areas tend to have bad schools with low funding; richer areas have much better schools with better funding. With magnet schools the state/school district/whatever will take a struggling school, pump a bunch of extra funding into it, and give it some sort of academic theme (e.g. visual arts, performing arts, STEM, etc.) with the goal of attracting gifted students from outside to come to the school.
Nor do you need an LLM - someone makes a cover letter once and then you use it as a template.