Chiang Rai has 6 main hospitals. Overbrook, a private christian hospital founded by missionaries about 100 years ago. Sriburin, a newer private hospital which is favored by most expats but very expensive, Inter Hospital, they like to push inflated priced medicines, and the newly opened Bangkok hospital. The government hospital is far more basic than the other two and much less expensive. If you have an emergency, it is recommended that the government hospital be your first port of call, especially if it is after hours as the private hospitals often have few if any doctors on hand at that time. Then we have the government teaching hospital at Mae Fah Luang University but the services there are limited.
As for the doctors, it makes no difference which hospital you choose because the same doctors rotate through all of the hospitals.
Depending on your needs, health insurance can cost anywhere between $1,000 per year for coverage in Thailand only to $3,000 plus per year for international coverage. Unless you are a frequent traveler, you probably should consider the cheaper option and purchase travel insurance as needed. We will be happy to connect you with someone who will take care of whatever you want health insurance wise.
You may have heard that you cannot buy insurance once you are past age 70 but that is untrue. I know of a health insurance company that will accept you at a cost of about 37,000 Baht per year for Thailand only coverage.
I am aware that most countries sell a health insurance policy that covers all family members but that is not so in Thailand.