Yes, it is an oversimplification overall. First, state laws vary about who can teach, and some laws apply to private schools as well. You may need certification or not to be hired, but in some places you will need to seek it afterwards. Most schools need to be certified by the state themselves, so the rules need to be followed.
Whether having been a TA or not may make little difference. What secondary school teachers do and what TAs do can be very different.
Nearly everywhere teachers need to do some kind of continuing education (to maintain certification). That might be in field, or it might be pedagogy, but it is usually a requirement. Having an advanced degree might alter the equation, but probably not erase the requirement.
Public schools aren't all unionized. Private schools don't all provide housing and few provide housing for everyone.
You probably don't need a doctorate anywhere in the US to be hired, but, some top secondary private schools value them.
Salary by region is probably a poor metric. If salaries are low, it is also likely that local living expenses are low as well. Some places excepted, of course, such as anywhere near the California Bay Area. But even New England isn't uniform. Boston is pretty expensive.
As for salary for an individual, the doctorate probably plays a relatively small part. The institution is looking for people who will serve their students well. Having a doctorate adds a bit of prestige to a school, but not necessarily an improvement in service to the students. They will look at the whole picture, as would any employer. Experience and letters of recommendation might weigh more than the degree.
But it is the complete picture and a judgement that you will fit in well that is more important than the degree.
After you are hired, your advance in salary depends little on the degrees you came in with. And most schools are constrained financially, both public and private.
