Local hospitals are cheap, international hospitals are not cheap at all. Have you ever been hospitalised at a local Thai hospital?
Maybe you are lucky that your home health insurance will cover parts of the emergency treatment. But you still have to pay out of own pocket, before you can be reimbursed in your home country.
I was under the impression that it was all language schools that had issues at the moment, and only official schools (like a uni) were still able to process the papers for a visa.
A little side note just FYI. The government is clamping down on language school/muay Thai EDs, and unless you study at a university, you might not get any extension in June.
I only mention this so it doesn't come as a surprise, and you will have little time to relocate your family.
Years ago it was not super hard to cheat the system, but it is getting harder and harder.
I managed 7 years in the country on EDs/Exempts and TVs, but 3 years ago I came to the end of the line and had to get a proper visa to be able to live here.
Living here on a short-term visa is a constant headache and not something I recommend. One day the party is over, and it is not fun if your whole life is in the country.