If you are going to talk about education, at least know what you are talking about. The writer sounds like he is a private school teacher, but his knowledge of public ed is something you can put on a head of a pin. He doesn't know what in the hell he is even talking about.
Private schools pay far less than public schools because those schools depend on tuition to operate, and tuition can't rise very much without worrying about parents taking their "business" elsewhere. Public schools have taxpayer funds, and they can "afford" to pay "more," though the pay for public school teachers is extremely inadequate.
Working conditions aren't necessarily better in private schools because the pay and benefits are crap. Almost all private school teachers go to work in the public sector because the pay and benefits are better. When there is a public school opening available, these "dedicated" private school teachers jump ship, and you can't blame them. If you have a rich spouse, you might actually "enjoy" working in a private school, but what this writer is saying is just nonsense. The teaching isn't really any different in private schools than in public. Teachers are in these schools because they can't get jobs in the public sector, so they try and get experience there to move onto the public sector. More than a few private school teachers are retired public school teachers, so the Ph.D. claim could very well apply to those retirees.
Furthermore, there is NOT a "supply" problem in public ed. You've got to be kidding when you make a claim like that. "Supply and demand" have no bearing on the public sector versus private sector. There may be more openings in public school jobs because the huge majority of teachers are in public ed, but there is a HUGE supply of newly-minted and unemployed teachers out here. It is nothing to find literally hundreds of applicants for a single public school district job, and thousands upon thousands of fully credentialed teachers, like yours truly, substitute for years and years on end for little money and NO benefits in an attempt to get a regular gig. Only in the least desirable school districts and schools are there any jobs to be had.
Teaching is notoriously difficult to break into, and, if you actually get a job in a public school, you have to worry about the increasing number of incompetent and insane principals who can destroy your career on a whim, unions and "tenure" notwithstanding.