No, It’s not competitive like musical theatre is competitive! That one is insane. No you don’t have to apply to 20 schools. My son applied to four.
Whether a prescreen is required depends on the school, even for music Ed. Prescreens are common for voice, including in music Ed. My son is a choral music Ed major at UNT and they require a prescreen for all voice applicants, including music Ed with a voice concentration. But not every school he applied to required a prescreen.
The best place to start looking for schools is the state where your student wants to teach. Make sure you understand the licensure requirements and any reciprocity or lack thereof between states for teacher certification. Ask school music teachers what state colleges put out the best music teachers—in our state there was a clear front runner for future music teachers. Then expand your search to the rest of the country, looking for schools that have great music Ed programs and will also result in a teacher certification that works for your preferred state.
We live in AZ, my son is getting his degree in Texas, which has a well-supported music education system in the public schools and some great teacher training. He is required to student teach in Texas. We have yet to see whether that makes it harder or easier to get a job in Arizona.