Exactly. Read again:
Different school environments make for radically different work, and many teachers find private schools offer a more rewarding experience. Attracting and retaining teachers, then, means more than just raising salaries. It means taking disciplinary obstacles and bureaucratic nonsense out of teachers’ paths.
The lesson, in short, is that you’ll attract more teachers by letting them teach.
There is less bureaucratic BS in the private school atmosphere than the public school atmosphere. This results in a better work environment for teachers, and they're willing to work for less. Ergo, it costs less to educate students in the private sector than it does in the public sector, because teachers are free of bureaucratic government BS and are allowed to do their jobs without everyone from voters to politicians second-guessing how they should approach the task of teaching their students.