In many places you do. We have what we call ‘open enrollment’ in the state of Colorado so you can go anywhere you can get in. The difficulty is that most of the more popular districts and schools give preference to those zoned to those schools (live in the catchment area), then have a list of priorities, like current students in the school, others in the school district, siblings. We have one district where you cannot opt into the main high school (ever) but most of the other high schools in the district do allow a few out of district students. Parents were renting apartments in the district and letting them sit empty to get their kids into the schools, and they even put an end to that (sent out investigators to see if they were actually living there).
Some schools also have special considerations, like a School of the Arts would have tryouts, or a magnet school might have an admissions test. The testing or auditions may take place in the school year before, so someone transferring into the area may not have the opportunity to get in. That happened to my kids when we moved to Florida and they missed the opportunities for the NEXT school year and weren’t even considered for the semester we moved. They also got last choice of classes for the current year and almost last for the following year.