As someone in the U.S., don't expect the timeliness to improve any with private healthcare.
In the state where I grew up, 18% of people have no health insurance at all.
There are some private practices that will let you pay a premium for quick appointments, but if you need a hospital, you're probably going to experience some wait.
https://www.oecd-ilibrary.org/sites/242e3c8c-en/1/3/2/index....