What you are looking for is a direct primary care provider. They do not accept insurance and therefore cut out all the overhead involved with insurance. They also cut out the obscene markup for diagnostic tests that traditional providers slather on.
https://duckduckgo.com/?q=direct+primary+care&t=brave&ia=web
Do you have experience with direct primary care? What was it like? Does the doctor spend a lot of time with you? Are blood tests covered by the membership?