Yes public funded health care has many issues, congratulations on being so astute. Where I live you can book longer appointments if you need them, you just have to actually ask for the extra time. People often have let small issues add up before getting them sorted because procrastination, small issues that most people with private healthcare systems could not afford to go to the doctor to have checked out.
Most important of all, when someone feels ill they don't have to factor a medical bill into the equation when deciding whether or no they should go to the hospital or possibly fucking die.