Trust me, you want a teacher who is busy— the good teachers are always busy, and the bad teachers are always available.
Good music teachers know that they are teaching life skills with many fringe benefits, that can inspire and build confidence, make friends fast, and are creative, among so many other things. When students find a teacher like this and realize it, they hold on to them forever.
Good teachers know that a great product (great teacher) is the best marketing plan, time management means efficient business, and that undercharging for their time adversely affects their clients in the form of spreading themselves too thin.
Bad music teachers think they are just teaching music, and at the slightest hint of adversity their clients will move on to a more worthwhile pursuit. They don’t hold students very long.
Bad music teachers think money and business are anti-art and anti-creativity, and don’t take their scheduling seriously. They don’t return phone calls or think of themselves as a business that requires customer service.
If you find a busy teacher, talk with them about when you can get started with them. It might be a week or two before they’re available but that’s ok and normal. In their industry they prioritize regular, longtime, consistent clients and the best ones they’ve had are the ones that were patient about getting started.