Maybe up north but south of I-80, this is absolutely no true.
Most HFC's are full time teachers or have admin positions at the school. However, most schools will not allow an admin (AD, Principal, Dean, etc.) to coach. I know of coaches who teach 6 and even 7 periods a day and have really good programs, perennial powerhouses, and even state champions. Not all are PE teachers either, some are even teach AP US History, College Level Math, and Dual Credit English.
Their stipends are probably a little above $5,000 but it varies from one district to the next.
If a school is paying $100,000 to a HFC, then their priorities are not about education, its about winning games. If you want to see a disadvantage of a public school, they cannot pay the HFC beyond what is stated in the Collective Bargaining Agreement, and as I said, its usually around $5000 or so. So they can't offer former NFL players, former college coaches, etc., large sums of money.
On a side note, I've seen people talk about private school coaches going to public schools and not having the same success. Are there any public school coaches that went to a private school and had more success?