That's because they cannot afford to do so. Their students are forced to pay not only private school tuition, but also pay public school tuition.
They do not even pay as much as their public school counterparts, and they know they don't have to, because teachers who teach in private schools are usually already fed up with public schools and their problems.
Well, one of the reasons that they can get teachers from public schools with less pay is that they don't have the same government-related UNsolveable problems. However, to draw teachers from the private sector, private schools DO have to pay. Thankfully, they can choose to pay whatever they want to whomever they want. Their hands are not tied by government-union negotiated pay scales. It is the case that public school teachers moving to a private school are always offerred less, for just the reason you state.
BTW, everyone who is a teacher knows that private schools pay LESS.
You don't have to be a teacher to know this. It is obvious that a business which prices its service at more than 5 times the cost of its competitor will have to cut costs where possible to stay in business-